2 Lecture Notes - Keele Universityrdj/ · Lecture notes a stray extra voltage in the circuit so that the actual voltage you are measuring is not the voltage you think you are measuring

2

Lecture Notes

Keele UniversitySchool of Chemical and Physical Sciences

1 Physics/Astrophysics LaboratoryLecture notes

2.1 The Laboratory Notebook

2.1.1 Introduction

Whenever a scientic experiment is performed the results of that experiment must berecorded in some way. However the record of an experiment is more than just a list ofnumerical measurements, it contains information on how, when, why and even where! you didthe experiment, what happened during the experiment, in fact all pertinent information. Forthe experimental strand of this module you are expected to record all such information in asingle book, your Laboratory Notebook.

Your laboratory notebook should be A4 or foolscap size with a sucient number of pagesto last for both semesters of the academic year. The book can be hardback or softback as youlike, however you should remember that hardback will stand up to any knocks it receives inthe laboratory better than a softback. At the end of each of your experiments your notebookwill be examined by one of the sta supervisors of the laboratory, who, based upon the con-tent of your notebook, will discuss your experiment with you and award you a mark for thatexperiment. Your notebook will also form your source of information when writing up yourlaboratory reports.

2.1.2 What Should You Record in Your Notebook?

The simple and glib answer to such a question is everything. This is unrealistic of coursebut you should make every eort to record as much as is practically and sensibly possible aboutyour experiment. It is not possible for anyone to predict the future and it is often very surpris-ing what information you may need to know at a later date.

An obvious thing you should record at the front of your notebook is your NAME, ADDRESSand that this is your SCHOOL OF CHEMICAL AND PHYSICAL SCIENCES notebook. Youshould take great care not to mislay your notebook but in the event that it is lost this can onlyincrease the probability of its recovery.

Each new experiment you start should be started on a new page in your notebook leavingone or two blank pages from your previous experiment. You should record the title of theexperiment and at the start of each laboratory session you should record the day and date atthe relevant point in the experimental record.

The laboratory notebook is by denition a note book, it is not expected to be a beautifullywritten piece of prose, quite the reverse is often the case. You should record events and resultsas they happen. However each of your notes should be in the form of a short sentence which issuciently explanatory that you or someone else can understand it.

The sort of information you record in your notebook will vary from experiment to experi-ment however some points that should be borne in mind in all cases are;

Record all of your data/measurements as you take them.



Always record the units of your measurements along with the measurements themselves.

If your data is taken for a certain period of time or a certain number of oscillations etc.then always record this fact along with the measurements themselves.

If you plot or t your data using one of the computer programs, make a printout ofthe program output (usually a graph) and attach (glue, sellotape, staple) it into yournotebook.

If you decide that a set of measurements is incorrect for some reason don't obliterate itin your notebook. Instead simply draw one diagonal line through it and make a note whyyou have discarded it. If at a later date you change your mind (or if a sta supervisoror post-graduate demonstrator persuades you to change your mind) you won't have tore-take the data again. As long as it can still be read it can be used.

Make a note of the pieces of apparatus that you are using in your experiment, e.g. ra-dioactive source B, A.C. circuit box G, a Farnell oscilloscope serial number F831GBXetc. If for some reason a piece of your apparatus is removed (it shouldn't be but!) thenwe can recover it if we know the number and you can continue your experiment withouthaving to start again.

At the end of your experiment you should summarise your results, tabulating clearly thevalues you have obtained for any derived quantities (and their error bars) with suitablenotes explaining what each is.

Your completed risk assessment form should also be attached.

2.2 Errors and the accuracy of experiments

2.2.1 Introduction

Every measured value of a physical quantity has an error associated with it and henceevery calculated result which is based on the measured values of physical quantities has anerror. In this context an error is not a mistake but the uncertainty in the value of a measuredor derived quantity. Whenever a value is quoted, especially as the result of an experiment, itserror bar should also be quoted. The error bar quanties the accuracy with which the valuehas been determined.

In any experiment there will be limits on how precisely or accurately we can measure aquantity. These limitations can arise from many dierent sources and in dierent ways. As anexample consider making a voltage measurement, some of the sources of uncertainty might be;

the smallest graduations on the scale of the voltmeter, (on a digital voltmeter this wouldbe the limit of the number of decimal places displayed)

a zero oset in the calibration of the voltmeter (analogue or digital)

electrical noise in the circuit causing uctuations in the voltage value



a stray extra voltage in the circuit so that the actual voltage you are measuring is notthe voltage you think you are measuring.

and lots of other possibilities as well. If you imagine almost any piece of experimental appa-ratus then with a little thought you can list a number of possible sources of uncertainty ina measurement. As an aside one of the most important tools in any experiment is a littlethought. Always consider carefully what your apparatus is actually doing, rather than anynumerical output it produces, in order to assess how accurately each part is working.

In the following sections we'll consider a number of aspects to do with errors (uncertain-ties) in experimental measurements. First in section(b) we'll consider how our uncertainty (orcertainty) in an experimental result aects how we can use that value in the scientic process.Then in section(c) we'll consider how errors in measurements can be classied and calculated.Following on from this in section(d) we'll write down some formulae for how we propagatethe eects of errors in a calculation. If we have a measured value we often want to put thatvalue into a formula to derive another value. Since our rst value contained some uncertaintyso too must the new value. Another situation is where we have two or more experimental valueseach with their uncertainty and we wish to combine (add, multiply,.. etc.) them together toget another value still. In section(e) there are some comments on the scale and importance oferrors in experiments.

2.2.2 The Relevance of Errors to the Scientic Process

If we have an experimentally measured value X for some quantity with an error bar ∆X(usually written in the format X ±∆X) then what does this mean? It means that we have a∼66 % condence that the true value of the quantity lies somewhere within the range X -∆X to X + ∆X and a ∼95% condence that it lies within X - 2 ∆X to X + 2 ∆X and a ∼99%condence that it lies within X - 3 ∆X to X + 3 ∆X. Thus experimentally we don't determinea specic value for a quantity but rather a range of values in which we believe that the truevalue lies. What happens if we want to compare our experimentally measured value X ±∆Xwith another value for the same quantity? This other value could be a value determined bya dierent method or by the same method but by a dierent person or it could be the resultof a theoretical calculation. Let's say (for example) that the other value is Y ±∆Y then thequestion that we ask is

does the value of X lie within the range Y - ∆Y to Y + ∆Yor

does the value of Y lie within the range X - ∆X to X + ∆X

If the answer to this question is YES then we say that the two values agree within thelimits of experimental error, otherwise they disagree. Two examples would be

i) A measured value of a resistance by Tom of 5 ± 1 Ω agrees within the limit of experimentalerror with the measured value of the same resistance of 4.7 ± 0.3 Ω made by Harry.

ii) The measured value of the density of the Sun of 1410 ± 10 kg m−3 agrees within experi-mental error with the theoretical value of 1407 kg m−3 predicted by the theory of Dr. Whobut disagrees with the theoretical value predicted by Mr Spock of 1457 kg m−3.



The conclusions we draw from the size of the error are not however just limited to agreementbetween two values their implication goes further. In example (i) above Harry's value has amuch smaller error bar than Tom's and we'd conclude that Harry's value was more precise thanTom's. In example (ii) we'd conclude that Dr. Who's theory was correct while Mr Spock'swas incorrect. Although if future experiments measured the mass of the Sun more precisely(smaller error bar) and Dr. Who's value then lay outside the error bar range we'd concludethat his theory too was now incorrect. What this would probably mean was that Dr. Who'stheory was not sophisticated enough to predict the mass of the Sun to the same accuracy withwhich it could be measured!

The interplay between experimental and theoretical physics revolves around this role ofexperiment to test the predictions of theory and of theory to accurately predict the results ofexperiments. In doing this, the error bar, the quantied accuracy of an experiment clearly playsa vital role.

2.2.3 Types of Error

The sources of errors in an experiment tend to be classied into one of two types, eitherrandom or systematic errors. In the next two subsections each of these two classes are discussedin more detail.

2.2.4 Random Errors

This class of error applies to the situations where a measured value uctuates randomlyabout its true value. Each time a measurement of the quantity is made a slightly dierentresult is obtained. An example used earlier was where electrical noise in a circuit causes a mea-sured voltage to vary. Another example is the nuclear beta particle decay process consideredin experiment A where the number of decays in a second uctuates randomly about a meanvalue. It is also possible for some measurements which do not possess an intrinsically randomnature to be eectively random. Imagine a situation where a large group of people are allasked to take a reading o of a scale. It is quite possible for dierent people, due to parallaxerrors, to read slightly dierent values o of the scale so that a random set of values uctuatingabout a mean is obtained.

A random error is one where in a series of measurements the values obtained are distributedin an unbiased and independent way about the true value. This type of error can be dealt withby repeating the measurement many times and then applying the ideas of statistical analysis.If we measure a quantity N times and obtain a set of values x1, x2, x3,.....,xN then we canestimate the true value for x from the mean value, i.e.

x =(x1 + x2 + x3 + ...xn)

N=

1

N

n∑i=1

xi (1)

The random values that we have measured were generated by a probability distributionfunction, which causes the underlying uctuation in the values. We usually consider that



for random independent errors this underlying probability function is a Gaussian or normaldistribution given by the equation

P (x) =1√2πσ

exp

(− 1

2

[x− xTσ

]2)(2)

where xT is the true value for x and σ is the standard deviation for the probability distribution.The standard deviation is the quantity that sets a scale for the range of uctuations that weobserve. We have estimated a value for xT from the mean of our measured values, we can alsoestimate a value for σ from our measurements by using the equation

σ =

√√√√ 1

N − 1

N∑i=1

(xi − x)2 (3)

However equation 3 is not our estimate of the error bar. This is because we know that there arerandom uctuations in the values of x and that our best estimate of the true value of xT wasthe mean. Our error bar is therefore a value for how accurately we have determined the meanvalue, not what is our estimate of the size of the uctuations. The formula for our estimate ofthe error bar ∆x is

∆x =σ√N

(4)

which is the error in the mean. If one considers equations 3 and 4 then one can see why theyare dierent. If we took an innite number of measurements we would be able to determine themean value of the distribution and hence the true value of x exactly and hence have ∆x = 0.However we would still have a distribution of x values, i.e. a non-zero value for σ, because ofthe underlying uctuations in the measured values of x.

It is obviously not possible to take an innite number of measurements of a quantity in anexperiment, however one should take enough measurements to ensure that ∆x is small enoughfor the purposes of an experiment. The formulae above give us a way of evaluating the mean,standard deviation and error bar, although nowadays it is more usually a case of simply enter-ing the set of values x1, x2, x3, ....., xN into our calculators and pressing the statistics buttons.The calculator then evaluates equations 1, 3 and 4 for us.

2.2.5 Systematic Errors

These are errors which are reproducible and which will be the same every time we repeata measurement. The most obvious type of systematic error is one of an incorrect calibrationof a piece of equipment. If for example the zero position on a voltmeter is incorrectly set andcorresponds to a small non-zero voltage then every voltage we measure will be shifted from itstrue value by this zero oset. Simply repeating the measurement many times will not remove,or allow us to calculate the size of, this inaccuracy.

In order to tell if a calibration error is present there are two things one might consider doing.One is to perform a calibration measurement of a standard value. For the example of our



voltmeter the calibration measurement would be to measure a xed known voltage and to seeif the voltmeter reading agreed with the standard value. Another approach to such a problemmight be to repeat the measurement with a dierent (independently calibrated) piece of equip-ment, or (eectively the same thing) to recalibrate the equipment independently and repeatthe measurement. The dierence between the two readings then gives an estimate of the sizeof the error. If one were to repeat this recalibration and measurement process many times thenone would eectively have turned a systematic error into a random error. In practice howevercalibration is usually a long process and one can only aord time for one or two measurements.

Another systematic error is the limit set by the graduations on a measuring scale. If forexample a ruler which is used to measure a length is graduated in 1mm steps then we cannotdetermine a length to better than 1mm and no matter how many times we repeat the measure-ment we we'll still only get a value accurate to 1mm. In section (2.3) there is a description ofhow the precision of measurement scales can be improved by the use of a vernier scale. Exactlythe same precision limit problem occurs with digital equipment, the limit of the display setsa lower limit on the accuracy with which a measurement can be made. This scale inaccuracygives an error value for ∆x.

2.2.6 The Propagation of Error Bars in Calculations

Once we have determined a value in an experiment we very often want to use that value ina formula to derive a further value. Since our experimental value has an error (an uncertainty)associated with it, it follows that our derived value will also be uncertain, i.e. have an error.How does the uncertainty in the derived value depend upon the uncertainty in the initial value?

2.2.7 The Propagation of Errors Through Functions

There is a general formula for propagating an error through a function in the limit that theerror bar is small. If we have a general function F(x) and a value X ± ∆X then

Z ±∆Z = F (X ±∆X) = F (X)±∆XdF

dx

∣∣∣∣x=X

(5)

In other words the error bar value is multiplied by the value of the derivative of the functionat point X.

Some specic examples are



(X ±∆X)2 = (X)2 ± 2X∆X

(X ±∆X)n = (X)n ± nXn−1∆X

sin(X ±∆X) = sin(X)±∆Xcos(X)

cos(X ±∆X) = cos(X)±∆Xsin(X)

loge(X ±∆X) = loge(X)± ∆X

X

log10(X ±∆X) = log10(X)±∆Xloge(10)

Xexp(X ±∆X) = exp(X)±∆Xexp(X)

10(X±∆X) = 10X ±∆Xloge(10)10X

2.2.8 Combining Errors

A slightly dierent problem of error propagation in a calculation is how do we nd thederived error when we combine (add, subtract, multiply or divide) two quantities where eachhas an error bar. Clearly the resulting combination will also have an error bar and withoutgoing into detail the equations for calculating this resultant error bar are;

Addition Z = X + Y ∆Z =√

(∆X)2 + (∆Y )2

Subtraction Z = X − Y ∆Z =√

(∆X)2 + (∆Y )2

Multiplication Z = X × Y ∆Z

Z=

√(∆X

X

)2

+

(∆Y

Y

)2

Division Z = X ÷ Y ∆Z

Z=

√(∆X

X

)2

+

(∆Y

Y

)2

2.2.9 Combining More than Two Quantities

What if we want to combine more than two quantities?, for example to multiply 3 quantitiestogether. The solution to this problem is to break the operation down into parts which onlyinvolve pairs of quantities and then just keep applying the rules above to each pair until oneobtains the nal answer. For example let us consider the equation

Z =XY

UVthen we break this down into the equations

Z =A

BA = XY B = UV

∆Z

Z=

√(∆A

A

)2

+(∆B

B

)2 ∆A

A=

√(∆X

X

)2

+(∆Y

Y

)2 ∆B

B=

√(∆U

U

)2

+(∆V

V

)2



which leads to the nal result

∆Z

Z=

√(∆X

X

)2

+(∆Y

Y

)2

+(∆U

U

)2

+(∆V

V

)2

2.2.10 Using Judgement About the Scale and Importance of Errors

In the previous sections we discussed how to calculate and how to propagate errors. Inherentin any good experimental practice is the desire to minimise the size of the error bar in the nalresult and hence to obtain as precise a value as is practically possible. Since any experimentmust be done in a limited amount of time we cannot reduce every error bar in the experimentto zero. Instead we must exercise judgement in deciding which are the important errors toreduce. In this context important means those errors which have the largest eect on the nalresult of our experiment. This means thinking about how that nal result is obtained and howeach source of error contributes to the error in the nal result and then minimising the biggestcontributor(s).

2.3 Scales and verniers

2.3.1 Introduction

The minimum separation between graduations on a scale sets a limit to the precision withwhich a reading can be made. A vernier can be used with a scale to improve the precision ofreading by a factor of 10 say.

Figure 1: A Vernier scale showing a measurement of 4.30 units.

Figure 1 shows a scale marked in centimetres and subdivided into units of 0.1 cm (i.e. mil-limetres). The vernier has a similar subdivision with markings at 0, 5 and 10. The 0 markindicates the desired reading on the scale but as indicated this can only be read easily to aprecision of ± 0.1 cm. In Figure 1 the mark 0 is lined up exactly with a subdivision of thescale, 4.2 cm. However the 10 mark on the vernier lines up exactly with the scale subdivisionat 5.1 cm, i.e. ten division of the vernier correspond with nine subdivision of the scale. Clearlythe markings on the vernier between 0 and 10 do not align exactly with any scale graduations.The interval between markings on the vernier is 0.09 cm; hence the rst vernier mark is 0.01 cm



short of a scale line (4.3 cm), the second mark is 0.02 cm short of the next scale line (4.4 cm) etc.

Figure 2: A Vernier scale showing a measurement of 4.33 units.

In the situation shown in Figure 2 the 0 mark is a little beyond 4.3 cm on the scale but thethird mark is exactly in line with a scale line (4.6 cm). To achieve this the vernier must havebeen moved 3 x 0.01 cm = 0.03 cm from its position in Figure 1, i.e. the scale and vernier arenow reading 4.33 cm.

To read a scale and vernier therefore we must read the scale graduation which is less than(or possibly equal to) the position of the 0 mark of the vernier. It also helps to estimate theposition of the 0 mark between the scale subdivision. The vernier mark which lines up with ascale line gives the next signicant gure for the measurement. It is not always easy to be surewhich mark on a vernier lines up exactly, or more closely, with a scale graduation. In these casesit often helps to (a) improve the illumination (b) use a low power magnier (c) look directlyalong the lines (A), not across them (B) (see arrows in Figure 2). Nevertheless there will alwaysbe some uncertainty (error) in the reading; usually at least ± 1

4of the smallest subdivision of

the vernier, often ± 1 such divisions and occasionally more, especially if the graduations arebroad, damaged, etc.

A scale and vernier may be used in a variety of situations and versions. A linear scale mayhave a subdivision of 0.1 but also have further subdivisions at the 0.05 values. The vernierwill then have a range 0 → 0.05 giving an improvement of ×2 in precision over the rst exam-ple. Particular care must be taken in reading the position of the 0 mark because a precisionof ± 0.005 is of little value if the position of the 0 is incorrectly read and there is an error of 0.05!

Circular scales which are used to measure angles (as on the optical spectrometer used inexperiment C) often have the scales graduated at intervals of 1° but with 20' subdivisions. Thevernier will then probably read 0 → 20' with subdivisions of 0.5'. Always be careful to checkwhere the 0 line is in this case it is easy to make mistakes of 20' or more in this case.



2.4 Spreadsheet and data analysis

2.4.1 Introduction

A spreadsheet is a program which allows us to manipulate sets of data. Such a programis contained in the MS-Excel package on the Physics Department PC's. Although spreadsheetscan be used for a number of purposes we'll just concentrate on one of the practical advantageswhich should be apparent to anyone who has had to calculate the errors in the gyroscopicmotion experiment! We'll use a spreadsheet to manipulate raw data values and their errors toobtain derived values and their errors. If, as in the gyroscope experiment, there are a largenumber of such values to be calculated a spreadsheet can provide a very ecient way to do this.

2.4.2 Basic points about a spreadsheet

1. Getting data into the spreadsheet

In Figure 3 an example spreadsheet is shown. After either starting MS-Excel, or equivalentspreadsheet program, the user is faced with a grid of the form shown in Figure 3. If the userhas selected an existing spreadsheet then the cells of the grid will probably already be lled inwhereas if the user selects a new spreadsheet the cells will be empty. Each cell can be referredto by its column letter and its row number, e.g. A3, C7, etc. The user can select a particularcell by using the mouse to click into that cell. Once in a cell one can perhaps more easilymove to other cells by using the up/down/left/right arrow keys. The current active cell is (a)highlighted by a bold border and (b) has its co-ordinates displayed in the leftmost box beneaththe toolbox line.

A B C D E F1 t (s) T (s) ∆t (s) ∆T (s) ωΩ ∆(ωΩ)2 0.355 11.40 0.005 0.05 9.755 0.1443 0.330 27.40 0.005 0.05 4.350 0.0664 0.380 33.30 0.005 0.05 3.120 0.0415 0.390 8.66 0.005 0.05 11.689 0.1646 0.275 15.85 0.005 0.05 9.057 0.1677 0.310 22.25 0.005 0.05 5.724 0.0938 0.375 6.50 0.005 0.05 16.196 0.2499 0.320 9.80 0.005 0.05 12.589 0.20710 0.230 23.00 0.005 0.05 7.463 0.16311 0.345 4.40 0.005 0.05 26.007 0.47912 0.290 9.60 0.005 0.05 14.180 0.255

Figure 3: Typical data set for gyroscopic motion experiment

In order to enter data into a cell simply type in the numbers (or text) that you want to put inthat cell. Once nished type a return (enter) and the data will be placed into the cell. You canedit (correct) the value in a cell by selecting that cell with mouse or arrows then clicking into



the cell contents box on the same line as the cell co-ordinates box. The left and right arrowsand delete keys can then be used along with the normal alpha-numeric keys to correct the value.

2. Manipulating cell values via simple formulae

Once the data is in the grid cells it can be manipulated. The usual procedure is to writethe answer from this manipulation into another grid cell. So for example if we want to multiplythe element in A2 by 2 say then (assuming column B is unused) we can click into cell B2 andthen in the cell contents box type = 2*A2 , the = is important since it tells the spreadsheetthat this is a formula. When we press return the value in B2 will be the numerical value of 2*A2.

What if we wanted to do this for all (or a range of) the values in column A and write theresults in column B ? Well we must rst do one element (e.g. B2 as above) and then use thell down command (which is under Edit in the bar at the top). First one must select the rstelement and also the other grid cells for which one wishes to calculate. This is down by clickingonto the rst cell/element with the mouse, holding the left bottom down and then draggingdown the elements of the column for which one wishes to calculate. Once this is done clickon Edit and select ll, and then down or up as appropriate. The values for each element areimmediately calculated, so that B3=2*A3, B4=2*A4, etc.

The example above was very simple and we can carry out more complicated operations. Wecan of course, add, subtract, multiply, divide ( symbols +, − , * , / ) by constants and alsoby other cell elements. For example if we select cell E2 we could enter in the contents box =A2+B2*(C2/D2)+3 which uses the values in columns A, B, C and D to calculate the values in E.

2.4.3 Function Names

We can also take functions of the cell values, for example if we select cell B2 then in MS-Excel in the contents box we can type = SQRT(A2) to take the square root or =LN(A2) to takethe natural logarithm. The following is a list of some of the MS-Excel spreadsheet functionswhich may be useful to you;

Function Syntax

Square root sqrt(a1)Raise to power N a1∧nNatural Log. ln(a1)

Log. to base 10 log(a1)Trig. functions sin(a1), cos(a1), tan(a1)Inv. trig. func. asin(a1), acos(a1), atan(a1)Sum a1 to a7 sum(a1:a7)

Mean of a1 to a7 average(a1:a7)Standard dev. of a1 to a7 stdev(a1:a7)



2.4.4 An Example The Gyroscope Data

In the gyroscopic motion experiment you measure the fast and slow periodic times for therotational motion (call these t and T) and from them must calculate the angular frequenciesω and Ω. Furthermore there are errors ∆t and ∆T in each of the measured times and whatone wants in the end is the product ωΩ and the error in this product ∆ωΩ. The formulae onewishes to use are therefore;

Ω =2π

TΩ

ω =2π

tω

∆Ω

Ω=

∆TΩ

TΩ

∆ω

ω=

∆tωtω

∆Ωω

Ωω=

√(∆Ω

Ω

)2

+

(∆ω

ω

)2

=

√(∆TΩ

TΩ

)2

+

(∆tωtω

)2

Let us assume that the t values are in column A, the T values in column B, the ∆t valuesin column C and the ∆T values in column D. Then in column E we can get the ωΩ values byselecting E2 and entering the formula

= 39.478 / ( A2 ∗ B2 )

which we can then ll down into the remaining cells in column E. The next step is to get theerror values. We can select F2 and use the following formula

= E2 ∗ SQRT( (C2/A2)∧2 + (D2/B2)∧2 )

which we can then again ll down into the remaining elements of the column. Hence we havethe values we wanted, worked out for us by the computer.

2.4.5 Saving, Updating and Printing

Save your spreadsheet regularly into a le, you don't want to have to type it all in againif you just need to change one value. If once you have completed your spreadsheet you ndthat a value is wrong (perhaps incorrectly entered) then when you change (edit) that value thespreadsheet is automatically updated and the later cells which depend on that value corrected.You can print o a copy of the spreadsheet by selecting PRINT under FILE.

2.5 The Linet program

2.5.1 Introduction

The Linet Program is a simple tting program written in Visual Basic to run underWindows for tting experimental data to either straight lines or to the AC circuit functionsused in the PHYSICS LABORATORY experiments in semester two. Linet is available on thePhysics Department PC's and can be started, either by double clicking on the Linet icon ifone exists, or by locating it in the programs list after clicking the windows start icon. Theplotting area of the Linet program usually lls the whole of the screen and so you may wantto maximise the Linet window once it has appeared.



2.5.2 Entering data into Linet

There are two scenarios, either this is the rst time data is to be entered or you are re-tting data already entered and saved onto disk. Either way rst click on File at the top ofthe window to get a drop down menu. If this is the rst time this data is being entered thenclick on Load Data and a data grid will appear with row labels, point 1, 2, 3, .... and columnlabels X-value, Y-value, Y-error. Use the mouse to click into a cell, e.g. point 1, X-value. Typeon the keyboard the number corresponding to this value, note it will only appear in the box atthe bottom of the grid, not in the cell itself. When the value is correctly entered (you can usedelete/back arrow etc. to edit the value) press return to record the value in the grid cell. Nowuse the mouse or up/down/left/right arrows to move to another cell and type in the value forthat cell, pressing return to load the value into the cell. Continue until all of your data pointshave been entered. Note : you must enter 3 values (x,y,y-error) for each point and you mustnot set the y-error to zero ! If you wish to go back and edit a value in the grid simply clickinto that cell with the mouse and type the new value in the box at the bottom of the grid,nishing with a return. When you are happy with the grid of data click on the OK button atthe bottom of the grid.

Once you've entered your data it is a (very) good idea to save it to your S: drive straightaway. Click on File and then on Save As in the drop down menu. Linet les should have theextension .dat and the standard MS-Windows Save window that appears has this extension pre-loaded. You can use the standard Windows disk/directory/lename method to select/constructthe location for your le. Once you're happy with the name click on OK to save the data to le.

If you are entering data from a previously saved le then you can do this by clicking on Fileand then clicking on Open from the drop down menu. The usual Windows Open File menuappears where you can select the disk/directory/lename of the le containing your data. Clickon OK when you have selected the correct le and the data will be loaded from the le. Noteif you now want to edit this data grid click on File and then on Load Data and the grid ofdata loaded from le will be displayed and can be edited as described before.

2.5.3 Plotting the data

Once loaded you can plot the data by clicking the Plot button at the bottom of the win-dow. If you want to set particular X and Y limits for your plot click on Limits at the topof the window. A drop down menu appears, click on the particular limit, or all 4 limits, thatyou wish to set and an input window appears. Type in the value(s) for the limit(s) into thisinput window and click on OK or press return. If you wish to change the X, Y or title la-bels for the plot click on Labels at the top of the window and select the particular label tochange, entering the new label into the input window in the same way as for the limits. In orderto update your plot with these new limits/labels you will need to click on the Plot button again.



2.5.4 Fitting the data

In order to t the data you have loaded you must rst select a function for tting, click onFunction at the top of the window. For straight line ts there are two choices, y=mx andy=mx+c in the drop down menu, click on whichever function is appropriate for your data toselect it. Now click on the Fit button at the bottom of the window and the t results willbe displayed in a box at the bottom right of the window. These values are m (the slope), c(the intercept), along with their error bar values, and χ2(the goodness of t parameter). Thedenition etc. of the goodness of t parameter and the least squares method used by the Linetprogram is described in section 2.8. In order to see the best t straight line plotted on top ofthe data you now need to click the Plot button again.

2.5.5 Printing the plot

In order to obtain a hardcopy of your plot click on File at the top of the window and thenon Print Plot in the drop down menu. This will activate the windows printing system ap-propriate for the particular PC you're using and you should answer the questions as per normal.

2.5.6 Transferring the data from Excel to LineFit

Highlight the data in the column containing your `x' axis values and select copy usingthe right hand mouse button.

Open a new Excel spreadsheet and paste the data into column A's second row usingpaste 123 or paste values option.

Highlight the data in column containing your `y' axis values and select copy using theright hand mouse button.

Paste the data into column B's second row using paste 123 or paste values option.

Highlight the data in column containing your `y' axis error values and select copy usingthe right hand mouse button.

Paste the data into column C's second row using paste 123 or paste values option.

In the rst cell in column A, enter the number od data points, i.e the number of rows inyour spreadsheet.

Save the le as Text(MS-DOS) le type using the save as option.

Use the LineFit program to open the above le and continue.

2.5.7 Exitting from the program

In order to exit from Linet click on File at the top of the window and then on Exit fromthe drop down menu. A conrmation window will appear, click on the OK button to exit theprogram.



2.5.8 Taking your own copy of Linet

If you have access to a PC running Windows you can take a copy of the Linet programfor your own use if you so wish. The Windows installation le can be downloaded from themodule webpage, available as a .zip le, save this le to your computer, open it by doubleclicking and then run the setup.exe le, this will install Linet to your computer. See Dr.A.Mahendrasingam, PHYSICS LABORATORY Module supervisor if you have any questions.

2.6 Preparation of laboratory reports

2.6.1 Introduction

In order to be of worth a piece of scientic research must be communicated to the rest of theworld. The method for doing this is by reporting the work in the scientic literature. In theelds covered by the fundamental sciences this is usually done by publishing a paper in one ofthe relevant scientic or technical journals which in Physics would for example be the Journalof Physics or Physical Review. For scientists working in a commercial environment their workwill have to be reported within the scientic management structure of their company. Theformat of such a report takes on the universally accepted style of a scientic paper.

∗ Abstract∗ Introduction∗ [Theoretical Background]∗ Experimental (or Theoretical) Method∗ Results∗ Discussion and/or ConclusionsThis scientic paper style is used when reporting the outcome of a single (or a small number

of) experiment(s). If a whole program of experiments is being reported then a dierent formatwill probably be used, which may be somewhat more akin to the structure of a book. Since inthis module you will only be tackling single experiments your laboratory reports should bewritten in this scientic paper format.

The following subsections discuss the various aspects of how you should write your labora-tory reports. In subsection 2.6.2 various general points pertinent to the whole of the report aregiven. Subsections 2.6.3(i - v) describe what are the expected contents of each of the sectionsof the report. Since you are expected to produce word processed laboratory reports subsection2.7 discusses various points associated with this.

2.6.2 General Points About Writing a Report

There are some general points about writing a laboratory report, or any technical document,which apply to the whole of the report and which are best discussed before more detailed pointsare considered in subsection 2.6.3.

i) All sections of the report must be written in proper English, it is not acceptable anywherein the report to simply use notes or just to have a collection of gures and tables.



ii) There should be a progressive logical ow in the content of the material presented in eachsection and in the report as a whole. In particular a reader should not have to searchahead for an explanation of a sentence or paragraph that they have just read or for thedenition of a symbol which has appeared for the rst time in the text.

In order to do this you would be well advised to make a plan of each section of your reportlisting, in the order that you wish to make them, the pertinent points for that section.Each new point you make should then correspond to a new paragraph in that section.Depending on the exact content of your report it may be sensible on occasion to go furtherthan this and to subdivide the section into subsections for dierent sub-topics dealt withwithin the section.

iii) Write in the third person, past tense;- e.g. The temperature was measured .... NOT Imeasured the temperature . . .

iv) Labelling Equations, Figures and Tables:- A particular feature of a technical documentis the inclusion of equations, gures and tables in the text. A gure could be either adiagram of apparatus or a graph of experimental results. All 3 of these types of includeditem should be labelled and in the text you should refer to them by those labels. The labelfor an equation is usually just a number at the end of the line on which the equation occurs.For a gure or table the label is usually on a line above or below the gure/table and thenumerical label is prexed by either the word Figure or the word Table as is appropriate.The numbering of equations, gures and tables should be sequential in the order thatthey appear in the text of the report. When referring to them in the text they should bereferred to for example as Equation(1), Figure(1) or Table(1). Examples of the labellingof equations, gures and table can be found throughout this manual.

v) Always quote the error bar when you quote a result in your text.

2.6.3 The Contents of the Sections of a Laboratory Report

i) AbstractThe abstract should contain a summary of the experiment, which is not more than ∼200words long. It should contain information on the purpose/aims of the experiment andthe results obtained. The abstracts from papers published in scientic journals are alsopublished separately (without the rest of the paper) in abstracting journals. These journalsof abstracts serve as a place where scientists can search to see if a paper contains informationthey are interested in and is worth reading in detail. The scientic journals themselvesnow publish the titles and abstracts of papers on the Internet so that people can downloadthe whole paper if they are interested in the particular work. It is therefore important inan abstract to present the most pertinent (and interesting ?) facts about your experiment,within the space allowed.

ii) IntroductionThis should explain the aim(s) and purpose(s) of your experiment and should put it intoperspective (i.e. present any relevant background material). Imagine that you are ex-plaining the aim/purpose to a physics student of your year of study who is not familiarwith the experiment and does not have a copy of the script for the experiment. The level



of presentation and the amount of detail should be appropriate to such a reader. Do notcopy substantial portions of the script for the experiment, the marker has read the scriptalready!! Your report should be written in your own words and not those of the scriptsauthor.

An outline of the theory should be presented. This should include a statement of the fun-damental Physics underlying the experiment (e.g. Newton's Laws, the wave theory of light,etc.), and any assumptions and approximations which have been made in the derivation ofthe equations you apply in the experiment. The only equations that appear in your reportshould be those that you use explicitly in your experimental analysis. If, however, there isa substantial amount of relevant theory then it is allowable, in fact preferable, to introducea new section altogether into the report, the Theoretical Background, which deals solelywith this material.

iii) ExperimentalThis section should describe your apparatus and experimental procedures. It is almostalways the case that you will nd it necessary to include a (schematic) diagram showingyour apparatus. Indicate the limitations of the apparatus both in terms of its range andaccuracy and the precautions taken to eliminate mistakes and to minimise uncertainties(errors). Suggest, where appropriate, how apparatus might be improved.

iv) ResultsThis section should describe and present the results of your experimental measurements.These results can often be classied into two types of result, raw data and derived results.

The actual measurements taken, e.g. the numbers of counts per second in the beta particleexperiment or the periodic times in the simple harmonic motion experiment are classiedas raw data. This raw data is usually presented in a graphical form, e.g. a plot of T2

against M in the simple harmonic motion experiment. If raw data is presented on a graphthen it should not (in general) be presented in a table also. On a graph the error bar foreach data point should be indicated by a vertical line drawn through the data point. Theline should extend +∆X above the point and ∆X below it where ∆X is the error bar.

The graphs of raw data presented should be clearly and properly labelled. It is conventionalto plot the measured y values as the vertical axis (ordinate) and the variable parameterx as the horizontal axis (abscissa). Each of the axes should have a label explaining whatthe variable plotted represents and what its units are. It is conventional to label an axis by;-

name of quantity (symbol) × power of ten unit

e.g. mass M × 10−3 kg

This is taken to mean that, for example, a value on the mass axis of 2.7 denotes a valueof 2.7 × 10 −3 kg. If there is not enough space on the axes for a clear explanation or iffurther explanation is required then a gure caption (a short paragraph of text) should be



added below the graph providing the necessary information.

A derived result is one that has been obtained by processing or analysing the set of raw datapoints. An example would be the value of the slope from a straight line t to the raw data.These results will normally just be quoted in the text of the results section. Only quote thederived result itself do not display any arithmetic performed in arriving at the derived result.If the derived result is taken from a graph then it should be clearly stated which gure itis taken from. All derived results should always have both their error bar and their unitsquoted with them.

v) Discussion/ConclusionComment on the signicance of your results. Has the aim of the experiment been achieved?If your aim was to verify a theory, over what range of values has this theory been veried?Did the results deviate from theory outside this range? Are the numerical results of theexpected order of magnitude? Do they agree with the generally accepted values (e.g. intables of physical data) within your estimated experimental uncertainty? Comment brieyon any important uncertainties of measurement and how they might be reduced.

2.6.4 Literature searching

Your report introduction will contain relevant background and theory, this will be achievedby searching and reading literature on the topic of your experiment, other than what is alreadyprovided in the experiment script.

Peer reviewed journals should be your prime source of information, websites can be a usefulstarting point in gathering information, however, use caution, they can often be inaccurate andcontain errors. Peer reviewed journals are reviewed by multiple people, which ensures that thequality of the information is to a certain standard and provides credibility to the methods andndings presented.

Web of Science, https://webofknowledge.com, is an excellent tool you can use to searchmultiple databases, most of the time you will have access to the full article, sometimes you willhave access only to the abstract, however it is still possible to get the full article through aninter-library loan.

Remember not only to use these references in your introduction, but also the rest of yourreport, journal articles and other references connect your own work with the wider physics com-munity and knowledge base. It also demonstrates you have understood the topic and relevanceof your work within the wider context.

2.7 Word processing of laboratory reports

2.7.1 Introduction

The two laboratory reports that you submit in this module must all be word processedand there are facilities for doing this available in the Physics Department, the University Li-



brary and at other locations around the University campus. If you have facilities of your ownfor producing word processed documents then please use those if you so wish, there is no par-ticular preference for any one or other word processing package.

2.7.2 Equations, gures and tables

In this manual the equations have been word processed and the gures and tables have beeninserted at the appropriate points in the text. Your word processed reports do not have to beas sophisticated as this. It is quite acceptable for you to write in equations and also specialsymbols (e.g. Greek letters etc.) by hand. There is however an equation editor available on thePhysics Department PC's and if you wish to use this facility it can only improve the presen-tation of your report. Figures and tables do not have to be inserted in the text of your reportbut can be attached at the back of the pages of text. Since they will be labelled as Figure(1)etc. and referred to as such in the text it should be clear to the reader which particular gureis being discussed. As an aside, when scientic papers are submitted to scientic journals forpublication, the journal companies in fact insist that gures are attached at the back of thetext rather than inserted.

2.7.3 Graphs and diagrams

Further to this your graphs and diagrams do not have to be created by the computer, handdrawn diagrams and graphs are acceptable. However in a number of experiments you are askedto t a straight line to your data and this can be done using the Linet program on the PhysicsDepartment PC's. In this case you should print o a copy of the resulting graph for inclusionin your report.

2.7.4 Report style

The text of your report should be in 12pt characters, in either Times, Times New Roman orCourier font. It should be double spaced and the pages should be numbered with your name onthe front page. Again, this is a format upon which the scientic journals insist when submittingpapers.

2.8 Fitting data to straight lines and least squares theory

2.8.1 Introduction

It is very common to display experimental raw data in a graphical form, whereby we plotthe measured values yi against the values of the independent variable xi which we have setin the experiment. In order to signify our condence (uncertainty) in the yi values we usuallyplot an error bar ±ei on the yi values. An example of such a plot is shown in Figure 4.



Figure 4: Example plot of experimental data

In many of the experiments described in this manual and in the case of the data plottedin Figure 4 we can intuitively see by looking at the data that we expect that underlying theraw data there is a straight line relationship between the values of y and x. The data points inFigure 4 don't lie on a perfect straight line however because of the eect of random errors. InFigure 4 a best t straight line has been drawn and again intuitively we'd say that we thoughtthis was a good description of the data. The question, of course, is how can we quantify ourintuition, how can we nd the straight line which best describes the relationship predicted bythe experimental data points and how can we tell whether this is a good description or not?In the following sections we'll describe the least-squares approach to these questions. This isby far and away the most commonly used method and is the algorithm used in the programsavailable on the PC's in the laboratory.

2.8.2 The Maximum Likelihood

In section 2.2.4 where we considered the eect of random errors we introduced the Gaussianprobability distribution for a measurement. Let us focus for the moment on just one of thedata points plotted in Figure 4, the i-th one. In the notation of this section, we have measureda value yi, the probability of whose measurement is given by

P (yi) =1√

2πσiexp

(− 1

2

[yi − yT (xi)

σi

]2)(6)

where yT (xi) = mxi + c is the true value at point xi, given by the straight line with the (as



yet unknown) slope and intercept m and c.

The probability of having measured the whole set of data points y1, y2, . . . . . . , yN is thereforegiven by multiplying together the probabilities for each of the y-values from equation 6, i.e.

P (y1, y2, . . . , yN) = P (y1)P (y2) . . . P (yN) ∝ exp

[− 1

2

N∑i=1

(yi − yT (xi)

σi

)2]

(7)

and is known as the likelihood of measuring the values y1, y2, . . . . . . , yN .

The probability in equation 7 depends upon 2, as yet unknown, values namely m and c. Inorder to dene our best t values for m and c we look at the probability given in equation7 from a slightly dierent angle. We ask the question what are the values of m and c forwhich it is most likely that we measured the set of values y1, y2, . . . . . . , yN? In a slightly moremathematical way, what are the values of m and c which maximise the likelihood?

2.8.3 The Least Squares Approach

The likelihood will be a maximum when the argument of the negative exponential in equa-tion 7 is a minimum. The minimisation of this argument is known as the method of least squares.In order to do this minimisation procedure we dene a goodness of t parameter given by;

χ2 =1

N − k

N∑i=1

(yi − yT (xi)

σi

)2

≈ 1

N − 2

N∑i=1

(yi −mxi − c

ei

)2

(8)

where the standard deviation σi has been estimated by ei and the co-ecient (1/N − k) is anormalisation term where N is the number of data points and k is the number of parametersestimated by tting the data (in this case k = 2 for m and c). It can be shown from statisticaltheory that the expected value of χ2 with this normalisation co-ecient is 1. This minimisa-tion of χ2 is equivalent to maximising the likelihood given in equation 7. The goodness of tparameter is a name which we've used to describe the role of χ2, it measures how good thet is, but it is also known by the more technical name normalised chi-square value and thequantity N − k is known as the number of degrees of freedom.

2.8.4 Minimising the Goodness of Fit Parameter

If we examine equation 8 then we can see that χ2 is only a function of the two parametersm and c since the values yi, xi and ei are the values measured in the experiment, not variables.A plot of χ2 against m and c would look something like Figure 5



Figure 5

The best t is when χ2 is a minimum and therefore mathematically the best t values for mand c must satisfy the equations

∂χ2

∂m= 0 = −2

N∑i=1

xi(yi −mxi − c)e2i

(9)

∂χ2

∂c= 0 = −2

N∑i=1

(yi −mxi − c)e2i

(10)

which can be re-written as two simultaneous equations in the unknowns m and c as

m

(∑i

x2i

e2i

)+ c

(∑i

xie2i

)=

(∑i

yixie2i

)(11)

m

(∑i

xie2i

)+ c

(∑i

1

e2i

)=

(∑i

yie2i

)(12)

which can be solved to get the values for m and c given by;

m =1

D

[(∑i

1

e2i

)(∑i

yixie2i

)−(∑

i

yie2i

)(∑i

xie2i

)](13)

c =1

D

[(∑i

yie2i

)(∑i

x2i

e2i

)−(∑

i

xie2i

)(∑i

yixie2i

)](14)

D =

[(∑i

1

e2i

)(∑i

x2i

e2i

)−(∑

i

xie2i

)](15)

Equations (13-15) give us a formula for determining the values of m and c which are thebest straight line t to the data. How good this t actually depends upon the value of χ2



with the best t values of m and c substituted. Remember since we have minimised χ2 this isthe lowest value that can be achieved by tting a straight line to this data. As noted earlierstatistical theory would suggest that if the data truly lies on a straight line but that thereare random errors in our measurements of the y values then we should have a value of χ2 ∼ 1.We can't expect to get a value for χ2 of exactly 1 in a real experiment but we'd hope to get avalue which is comparable to 1. In any report we make we can always quote the value of χ2 asa measure of how well a straight line describes our experimental data.

Since our data points have error bars (i.e. uncertainties) so too must our values for m andc. These error bar values are given by the equations;

∆m =

√1

D

(∑i

1

e2i

)(16)

∆c =

√1

D

(∑i

x2i

e2i

)(17)

In practical use we always evaluate equations (13-15) and (16-17) through a computer pro-gram. We enter the data into the program and the program then evaluates the equations andusually plots a graph as well as printing out the values of m, ∆m, c and ∆c.



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