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Introduction to the Physics Laboratory

In this laboratory you’ll do experiments that illustrate certain physical

phenomena and give a glimpse of the experimental data supporting physical

laws. I hope you will also experience some of the thrill of discovery that isthe essence of physics.


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Error Analysis and Graphs

Experimental data contains uncertainties, and in computing results from

them we wish to preserve the highest honestly allowable precision. Such

calculated results should give both the value found the degree of uncertaintyis this resultant value. Many methods achieving this are found in many

texts; a brief review of some of the simpler techniues is given below.

Types of Errors

Errors may be classified into two !inds, systematic and random.

Systematic errors are li!ely to be constant and in the same direction. "hey

are caused by faults in the apparatus or flaws in the observer’s techniue. It

is therefore freuently possible to discover them and reduce their

magnitudes.

#s an example where systematic errors would influence a measurement,

consider the measurement of length using a meter stic!. If the meter stic!

had a distorted scale because it was badly made, or its length varied with

environmental humidity or temperature, we would incur a systematic error in

our length measurements if we were unaware of these facts.

$andom errors are those errors that are produced by unpredictable and

un!nown variations in the total experimental process even when one doesthe experiment as carefully as is humanly possible. "hese could be due to

fluctuations in the line voltage, temperature changes, mechanical vibrations,

or any of the many physical variations that may be inherent in the euipment

or any other aspect of the measurement process. "he good news with

random errors, as opposed to systematic errors, is that they can be dealt with

in a consistent, statistical fashion. If enough measurements are ta!en, a

histogram of the results should loo! li!e a bell%shaped curve &or '(aussian

curve)* with the mean, or true, value at the pea! of the curve.

Accuracy and Precision

"he central point to experimental physical science is the measurement of

physical uantities. It is assumed that there exists a true value for any

physical uantity, and the measurement process is an attempt to discover that

true value. +n the other hand, it is not assumed that the process will be

perfect and lead to the exact true value. Instead, it is expected that there will


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be some difference between the true value and the measured value. "he

terms accuracy and precision are used to describe different aspects of the

difference between them. rom a scientific point of view, these have very

different meanings.

"he accuracy of a measurement is determined by how close the result of the

measurement is to the true value. or example, several experiments

determine a value for the acceleration due to gravity. or this case the

accuracy of the result is decided by how close it is to the true value of -./

m0s1. or several of the laboratory experiments, though, the true value of the

measured uantity is not !nown and the accuracy of the experiment cannot

be determined from the available data.

"he precision of a measurement refers essentially to how many significant

digits there are in the result. It is an indication also of how reproducible theresults are when measurements of some uantity are repeated. 2hen

repeated measurements of some uantity are made, the mean of those

measurements is considered to be the best estimate of the true value. "he

smaller the variation of the individual measurements from the mean, the

more precise the uoted value of the mean is considered to be. "his idea

about the relationship between the si3e of the variations from the mean and

the precision of the measurement shall be elaborated later.

4et’s loo! at the following table of values acceleration due to gravity

measured by four students5

#lf dev 6eth dev 7arl dev 8ee dev

Measurement

9

:. 9. -.:1 /./>

Measurement

1

99. -.1< .:> -.:<

dev ? deviation from the mean value


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@otice by the definitions of accuracy and precision that 8ee’s value of -.:<

is the most accurate while 7arl’s is the least accurate, and 7arl’s value is the

most precise while #lf’s is the least precise.

@otice the interplay between the concepts of accuracy and precision that

must be considered. If a measurement appears to be very accurate, but the

precision is poor, the uestion arises whether or not the results are really

meaningful. 7onsider #lf’s mean of -.>, which differs from the true value

of -./ by only /.: and thus appears to be uite accurate. Aowever, all of

his measurements have deviations greater than /.:, and two of his

deviations are much larger than /.:. It seems much more li!ely than that

#lf’s mean of -.> is due to luc! than to a careful measurement. If seems

li!ely, however, that 8ee’s mean of -.:< is meaningful because the

deviations of her individual measurements from the mean are small. In other

words, unless a measurement has high precision it cannot really beconsidered to be accurate.

#n examination of the significant figures given in this data leads to

essentially the same evaluation of each student’s data. 7onsider #lf’s data,

which indicates by the values stated for the individual measurements that

two places to the right the decimal point are significant. Aowever, that

conclusion is not supported by the fact that his deviations occur in the first

digit to the left of the decimal point. +n the other hand, 8ee’s results show

deviations in the second place to the right of the decimal point in agreement

with the fact that two places to the right of the decimal are given as

significant in the measured values. "hus from another point of view, 8ee’s

results are seen as meaningful, but #lf’s are uestionable.

7arl’s results, on the other hand, are an example of a situation that is

common the interplay between accuracy and precision. 7arl’s precision is

extremely high yet his accuracy is not very good. 2hen a measurement has

high precision but poor accuracy, it is often the sign of a systematic error. #

systematic error is an error that tends to be in the same direction for repeated

measurements, giving results that are either consistently above the true valueor consistently below the true value. In many cases such errors are caused

by some flaw in the experimental apparatus, li!e not calibrating a device

correctly. #nother source of a systematic error is failing to ta!e into account

all of the variables that are important in the experiment. or 7arl, if all his

value were consistently below the true value, this might represent 7arl

forgetting to ta!e into account friction, which would indeed cause all his


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values to be low. 6ut since his mean is well above the true value, this points

to a systematic error involving the euipment.

Percent Error and Difference

In several laboratories, the true value of the uantity being measured will be

!nown. In those cases, the accuracy of the experiment will be determined by

comparing the best estimate of the true value, or experimental value, with

the !nown true value. "his can be done by figuring the percentage deviation

from the !nown true value &as !nown as percentage error*. If E stands for

experimental value, and B stands for the !nown value, then5

In other cases a given uantity will be measured by two different methods.

"here will then be two different experimental values, E9 and E1, but the true

value may not be !nown. or this case the percentage difference between

the two experimental values can be calculated, but note that this tells nothing

about the accuracy of the experiment, but should be a measure of the precision. "he percentage difference is defined by5

Measurement and Significant Figures

4et’s say that you need to measure the length of a piece of string with a

meter stic!. "he meter stic! in uestion has as its smallest mar!ings a

C9// x K K E error Percentage−

=

C9//

191

91 x E E

E E difference Percentage

+

−=


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centimeter. Dou measure the length and find that it falls about halfway 1:

and 1 cm. Dou estimate that the length is 1:.= cm, but the /.= cm is not

exact, but a guess, so you could report that the length of the string to be

1:.= ± /.9 cm. 2hen a number is reported, typically the number of digits

reported is the number !nown with any certainty. "he uncertainty isgenerally assumed to be one or two units of the last digit, but may be

different depending on the situation.

2hen counting the number of significant figures5

• #ll digits 9 through - count as significant figures• eroes to the left of all of the other digits are not significant• eroes between digits are significant• eroes to the right of all other digits are significant if after the decimal

point and may or may not be significant if before the decimal point.

or example,

@umber @umber of Significant

igures

Fossible $ange of the $eal

Measurement

9.1 1 9.9 % 9.

.


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sing exponential notation helps removes this ambiguity. or example5

x 9/9 means &±9* x 9/9 ? measurement between :/ and -/

./ x 9/

9

means ./ &J /.9* x 9/

9

? measurement between :- and 9.// x 9/9 means .// &J /./9* x 9/9 ? measurement between :-.- and /.9

+ften you will have to mathematically combine various measurement values

with significant figures. "o ensure that the result represents the proper value

with the correct amount of uncertainty, these rules should be followed5

9* 2hen adding or subtracting, figures to the right of the last column in

which all figures are significant should be dropped.

1* 2hen multiplying or dividing, retain only as many significant figures

in the result as are contained in the measurement value with the leastnumber of significant figures.

* "he last significant figure is increased by 9 if the figure beyond it

&which is dropped* is = or greater.

"hese rules apply only to the determination of the number of significant

figures in the final result.

or example5 :=.9 1:.1

:./ x


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@ow, dx is to be associated with our uncertainty in x, and dy with our

uncertainty in y. df will then be associated with our uncertainty in f resulting

from dx and dy. @ow a difficulty arises. 2e do not !now whether dx and

dy are positive or negative uantities since we have assumed the

uncertainties to be due to the randomness of nature. Since it is generally

best to assume the worst, both uncertainties are ta!en to be in the same

direction; hence

where the bars denote absolute value, and where we have changed our

notation to reflect the fact that our uncertainties are not true differentials,

i.e., we have made the identifications

"he uncertainty would be given by

and we would write5

>9-.- ± /.: cm1

Aere are some formulas for finding the uncertainties in functions ofindependent variables whose uncertainties are !nown5

unctions ncertainty formula

@ote that this includes the commonly occurring cases of x1 and x/.=

ydy xdx f df ∆→∆→∆→ ,,

x y y x f ∆⋅+∆⋅=∆ max

1

max :.///=./;.:;//=./1:.


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Aowever, if f&x, y* is a complicated function of x and y, it is generally easierto proceed by calculating f / ? f&x,y*, f 9? f&xL∆x*, yL∆y* and f 1 ? f&x L ∆x, y

G ∆y*, where x and y are the measured values. &"he important point is that∆x and ∆y occur in f 9 with the same sign, whereas in f 1 they occur withopposite signs.* "he uncertainty in f&x,y* is then given by the larger of the

two magnitudes f / G f 9 and f 9 G f 1.

If you !now calculus, you can Nust determine the differential form of the

euation, and then divide that by the original euation.

The Mean and The Standard Deviation for Repeated Measurements

or a set of measurements where it is assumed that only random errors are

present, the mean value you determine represents your own best estimate for

the true value for whatever it is you are measuring. "he mean is given by

the euation5

or example, assume four measurements are made of some uantity x, and

the four results are 9.. 6y the above euation, the

mean value is5

Statistical theory, furthermore, states that the precision of the measurement

can be determined by calculation of a uantity called the 'standard

deviation) from the mean of the measurements. "he symbol for standard

deviation from the mean is σ. In a statistical sense, it gives the probabilitythat the measurements fall within a certain range of the measured mean and

is defined by the euation5

or the data given, the standard deviation is calculated to be5

∑

=

n

i xn

x9

9

( ) /.9->.1/:.9:;.9-

9 =+++

= x

[ ]∑ −

−=

n

i x x

n 9

1

9

9σ

x x f

x x f

cos*&

sin*&

=

=

x x f

x x f

∆=∆

∆=∆

*&sin

*&cos


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Frobability theory states that approximately


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a perfect correlation, and $ 1? / signifying no correlation at all. "hus, if the

computer calculates an $ 1 ? /.--, this means the data almost exacts lies

along the least suares fit line; while if the computer calculates r ? /.//1

means that the data points on your graph probably loo! li!e a shotgun

pattern and the least suares fit line pretty much means nothing and that

your data is not actually linear at all.

Preparing "raphs

# graph is often a useful way to represent data. "his section outlines some

of the main considerations in drawing a graph.

Each graph should have a title, e.g. '8istance "raveled as a unction of

"ime), or simply 'v vs. t) if v and t are defined in the report. 8o not forget

to label the coordinate axes with the variables and their units. 7hoose scalesfor the axes which will spread the experiment points over the entire graph.

(enerally, it is best to show the origin, but not if the data occupies a narrow

range of values far from the origin. 7onventionally, the independent

variable is chosen for the abscissa &hori3ontal axis* and the dependent

variable for the ordinate &vertical axis*. "he title traditionally is written as

'what is on the ordinate) vs. 'what is on the abscissa.)

If possible, draw uncertainty 'bars) on each of your data points. "hese

uncertainty bars are the graphical representation of the uncertainty in your

data.

+ften it is very helpful to fit a line or curve through your data points. "he

software programs Excel and +rigin :./ do an excellent Nob of this. 2hen

this is done, it is important to ma!e sure that euation for the fit is displayed

on the graph as well as the some sort of uality of fit statistic, li!e $ 1 &see

above*. or the euation of the fit, it is important to change the variables to

those displayed on your graph.

4et’s us consider an example. "he velocity of a ball undergoing constant

acceleration has been measured as a function of time. "he data are5

"ime &sec* Pelocity &m0s*

9./ ./

1./ .:

./ >.>

>./


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2e will assume the uncertainty for all data points to be ±/.9.

"he graph &using Excel* is shown below5

+ccasionally it is helpful and0or necessary to extend the line or curve

beyond the range of measurements. "his is called an extrapolation and

should be indicated by a dotted line&s*.

Velocity of the Ball as a Function of Time

v = 0.7514t + 2.2086

R2 = 0.9986

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

0 1 2 3 4 5 6

Time (sec)

V e l o c i t y ( m / s )


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Laboratory Reports

"he following paragraphs will give you an idea of what a report shouldcontain and what it should accomplish. "he order in which the contents

appear is of secondary importance, but they must all be there, clearly stated

and in a manner suited to the particular situation.

"he purpose of a lab report such as this is to communicate fully and

coherently the results of an experimental investigation. Fersons outside the

immediate group performing the experimental wor! may be interested in the

results of your wor!. It is therefore important that these results and your

analysis of them be put forth in a form suitable for such communication."he effectiveness of this communication will depend on the clarity and

conciseness of your explanation of the wor! and the facility with which

desired material can be extracted from it.

#lthough there is no set universal form for reporting laboratory wor!, due to

the varied nature of experimentation, there are a few basic things which all

technical reports on experimental wor! should do. # lab report should at the

very beginning tell the reader what the experiment was for; what it

attempted to find out. Secondly, the report should inform the reader, in a

concise way, how the wor! performed is expected to accomplish this purpose. "hirdly, where warranted, the report should display calculations

and analysis, charts, and figures, and at times significant raw data. ourthly,

and most importantly, it should inform the reader of the particular results,

their significance, conclusions which may be drawn from them, and the

Nustification of such conclusions.

"here are in general five maNor parts to a report even if a formal distinction

is not made between them. "hese are as follows5

9. Objective5 "he abstract tells the reader, when he0she first starts

reading, what he0she will find in the report if he0she reads it through.

In one or two sentences it should summari3e all topics presented in the

report.


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1. Introduction/Teory5 "he introduction gives the purpose of the

experiment; it also indicates what principles are used, and how, in

order to obtain the results sought.

. Ex!erimenta" Procedure5 "he procedure &which is often combined

with the introduction* gives either a statement of standard methods

used, or if the experimental method is of particular interest, a brief

description of that method. 8o not repeat material included in the

descriptive writings distributed for the experiment.

>. #esu"ts/$na"ysis5 Dou should always present5

9* sample calculations to show how you got any derived results;

1* figures and tables representing results either final orintermediate; and

* the significant data upon which your wor! was based.

$epetitive calculations and arithmetic should not be shown, but

samples of repeated calculations and any other calculations of

interest should be presented in such a way that it is clear to any

competent reader Nust what you are calculating, how you are

calculating it, and what the answer is.

=. %iscussion/&onc"usion5 It is important that final results and any

intermediate results of importance be explicitly presented in the

written body of the discussion. "he degree to which the intermediate

results should be included is a matter of Nudgment on the part of the

author. It is up to him0her to consider how well the report answers

any uestions which might arise in the reader’s mind.

Some analysis and interpretation of your results must be included to

give them meaning. "heir significance to the immediate purpose

should be shown clearly; and if they warrant it, one may include therelationship of these results to the field in general.

7ertainly any report of experimental results must contain an

indication and0or discussion of the reliability of the results, including

a uantitative statement thereof. "hese reliability statements,


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combined with theoretical considerations, form a basis for the

Nustification of your conclusions.

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