Section 1 – Treatment of uncertainties in Physics at AS and A2 level Preamle
One of the main aims of the practical work undertaken in GCE Physics is for candidates to develop a feeling for uncertainty in scientific data. Some of the treatment that follows may appear daunting. That is not the intention. The estimates of uncertainties that are required in this this spec specifi ifica catio tion n are more more in the the natu nature re of educ educat ated ed guess guesses es than than stati statist stica icall lly y sound sound calculations. t is the intention that candidates !e introduced early in the course to estimating uncertainties so that !y the time their work is assessed" they have a rela#ed attitude to it. The sections in P$% on density determinations and resistivity are ideal for this. !efinitions "ncertainty &ncertainty in measurements is unavoida!le and estimates the range within which the answer is like likely ly to lie. This This is usua usuall lly y e#pr e#press essed ed as an a!sol a!solut utee valu value" e" !ut !ut can !e give given n as a percentage. The normal way of e#pressing a measurement x measurement x'" with its uncertainty" u" is x is x' ( u. This means that the true value of the measurement is likely to lie in the range x range x' − u to x to x' ) u.
*ote+ The term ,error- is used in many te#t!ooks instead of uncertainty. This term implies that something has gone wrong and is therefore !est avoided. &ncertainties can !e split up into two different categories+
-
-
#andom uncertainties These occur in any measured quantity. The uncertainty of each reading cannot !e reduced !y repeat measurement !ut the more measurements which are taken" the closer the mean value of the measurements measurements is likely to !e to the ,true- value of the quantity. Taking repeat readings is therefore a way of reducing the effect of of random uncertainties. Systematic uncertainties uncertainties These can !e due to a fault in the equipment" or design of the e#periment e.g. possi!le /ero error such as not taking into account the resistance of the leads when measuring the resistance of an electrical component or use of a ruler at a different temperature from the one at which it is cali!rated. The effect of these cannot !e reduced !y taking repeat readings. f a systematic uncertainty is suspected" it must !e tackled either !y a redesign of the e#perimental technique or theoretical anal analys ysis. is. 0n e#am e#ampl plee of this this sort sort of uncer uncerta tain inty ty"" the the orig origin in of whic which h rema remain inss mysterious" is in the determination of stellar distances !y paralla#. The differences !etween the distances" as determined !y different o!servatories" often e#ceeds the standard uncertainties !y a large margin.
Percenta$e uncertainty This is the a!solute uncertainty e#pressed as a percentage of the !est estimate of the true value of the quantity. #esolution This is the smallest quantity to which an instrument can measure
%istake This is the misreading of a scale or faulty equipment. Suspect results These are results that lie well outside the normal range e.g. points well away from a line or curve of !est fit. They often arise from mistakes in measurement. These should !e recorded and reason for discarding noted !y the candidate. &ow is the uncertainty in the measurement of a 'uantity estimated( 1. )stimation of uncertainty usin$ the spread of repeat readin$s. Suppose the value a quantity x is measured several times and a series of different values o!tained+ x%" x1" x233.. xn. 4*ormally" in our work" n will !e a small num!er" say 2 or 56.
&nless there is reason to suspect that one of the results is seriously out 4i.e. it is anomalous6" the !est estimate of the true value of x is the arithmetic mean of the readings+ x% + x1 + ........xn n 0 reasona!le estimate of the uncertainty is 8 the range+ 7ean value x =
xma# − xmin " where xma# is the ma#imum and xmin the minimum reading of 1 x 4ignoring any anomalous readings6 i.e.
u=
)xample The following results were o!tained for the time it took for an o!9ect to roll down a slope. :.5 s" :.; s" :.< s" 5.% s" 5.' s
The !est estimate of the true time is given !y the mean which is+ t =
:.5 + :.; + :.< + 5.% + 5.' 5
The uncertainty" u" is given !y+ u =
= :.;s
5.% − :.5 = '.2s 1
The final answer and uncertainty should !e quoted" with units" to the same no. of decimal places and the uncertainty to % sig. fig i.e. t = :.; ( '.2 s *ote that" even if the initial results had !e taken to the nearest '.'% s" i.e. the resolution of an electronic stopwatch" the final result would still !e given to '.% s !ecause the first significant figure in the uncertainty is in the first place after the decimal point. The percentage uncertainty" p
=
'.2 :.;
× %''> = <>
. 0gain" p is only e#pressed to % s.f.
2. )stimation of uncertainty from a sin$le readin$
Sometimes there may only !e a single reading. Sometimes all the readings may !e identical. Clearly it cannot !e therefore assumed that there is /ero uncertainty in the reading?s@. Aith analogue instruments" it is not e#pected that interpolated readings will !e taken !etween divisions ?this is clearly not possi!le with digital instrument anyway@. $ence" the uncertainty cannot !e less than 8 the smallest division of the instrument !eing used" and is recommended it !e taken to !e ( the smallest division. n some cases" however" it will !e larger than this due to other uncertainties such as reaction time 4see later6 and manufacturerBs uncertainties. f other sources of random uncertainty are present" it is e#pected that in most cases repeat readings would !e taken and the uncertainty estimated from the spread as a!ove.
Advice for Specific apparatus %etre #ule
Take the resolution as (% mm. This may !e unduly pessimistic" especially if care is taken to avoid paralla# errors. t should !e remem!ered that all length measurements using rules actually involve two readings one at each end !oth of which are su!9ect to uncertainty. n many cases the uncertainty may !e greater than this due to the difficulty in measuring the required quantity" for e#ample due to paralla# or due to the speed needed to take the reading e.g. re!ound of a !all" in which case the precision could !e ( % cm. n cases involving transient readings" it is e#pected that repeats are taken rather than relying on a guess as to the uncertainty. Standard %asses
or 1'g" 5'g" %''g masses the precision can !e taken as !eing as !eing (%g this is pro!a!ly more accurate than the manufacturerBs 4often a!out 2>6. 0lternatively" if known" the manufacturerBs uncertainty can !e used. !i$ital meters *ammeters+voltmeters,
The uncertainty can !e taken as !eing ( the smallest measura!le division. Strictly this is often too accurate as manufacturers will quote as !igger uncertainty. 4e.g. 1> ) 1 divisions6 Thermometers
Standard D%' C to %%' C take precision as %C Figital thermometers uncertainty could !e ( '.%C. $owever the actual uncertainty may !e greater due to difficulty in reading a digital scale as an o!9ect is !eing heated or cooled" when the su!stance is not in thermal equili!rium with itself let alone with the thermometer..
The period of oscillation of a Pendulum+Sprin$
The resolution of a stop watch" used for measuring a period" is usually '.'%s. eaction time would increase the uncertainty and" although in making measurements on oscillating quantities it is possi!le to anticipate" the uncertainty derived from repeat readings is likely to !e of the order of '.% s. To increase accuracy" often %' ?or 1'@ oscillations are measured. The a!solute error in the period 4i.e. time for a single oscillation6 is then %H%' ?or %H1' respectively@ of the a!solute error in the time for %' ?1'@ oscillations e.g. 1' oscillations+ Time = %5.; ( '.% s 4'.<>6 %5.; ± '.% ∴ Period = s = '.IJ' ( '.''5 s 1' *ote that the percentage uncertainty" p" in the period is the same as that in the overall time. n this case" p =
Smallest measura!le quantity e.g. ( % cmK" !ut this depends upon the scale of the instrument. n the case of measuring the volume using the line on a !eaker" the estimated uncertainty is likely to !e much greater. *ote candidates must !e careful to avoid paralla# when taking these measurements" and should state that all readings were taken at eye level. They should also measure to the !ottom of the meniscus.
!eterminin$ the uncertainties in derived 'uantities. Please note that candidates entered for AS award will now e re'uired to comine percenta$e uncertainties.
Lery frequently in Physics" the values of two or more quantities are measured and then these are com!ined to determine another quantityM e.g. the density of a material is determined using the equation+ m ρ = V To do this the mass" m" and the volume" V " are first measured. Each has its own estimated uncertainty and these must !e com!ined to produce an estimated uncertainty in the density. The volume itself may have !een determined !y com!ining several independent quantity determinations 4e.g. length" !readth and height for a rectangular solid or length and diameter for a cylindrical wire6. n most cases" quantities are com!ined either !y multiplying or dividing and this will !e 2 considered first. 7ultiplying !y a constant" squaring ?e.g. in 2: π r @" square rooting or raising to some other power are special cases of this and will !e considered ne#t. 1. %ultiplyin$ and dividin$The percentage uncertainty in a quantity" formed when two or more quantities are com!ined !y either multiplication or division" is the sum of the uncertainties in the quantities which are com!ined. )xample
The following results were o!tained when measuring the surface area of a glass !lock with a 2'cm rule" resolution '.%cm Nength = J.I ( '.% cm Aidth = :.: ( '.%cm *ote that these uncertainties are estimates from the resolution of the rule. This gives the following percentage errors+ '.% × %''> = %.'> J.I '.% pA = × %''> = 1.1> Aidth :.: So the percentage error in the volume" pL Nength+
pN
=
= %.' + 1.1 = 2.1>
$ence surface area = J.I × :.: = :1.<; cm ( 2.1 > The a!solute error in the surface area is now 2.1> of :1.<; = %.2I cm uoted to % sig. fig. the uncertainty !ecomes % cm The correct result" then" is :2 ( %cm D *ote that surface area is e#pressed to a num!er of significant figures which fits with the estimated uncertainty. 1. #aisin$ to a power e$ x 2/ x 1/
x
0
The percentage uncertainty in xn is n times the percentage uncertainty in x. e.g. a period ?T @ is as !eing 2% seconds with a percentage uncertainty of 1 >"
So T 1 = J<% ( :>. :> × J<% = :' ?to %.s.f@ So the period is e#pressed as T = J<' ( :' s. −
*ote+ x % is the same as %H x. So the percentage uncertainty in %H x is the same as that in x. Can you see why we ignore the − signQ *ote+ the percentage uncertainty in x is half the percentage uncertainty in x. . %ultiplyin$ y a constant n this case the percentage uncertainty is unchanged. So the percentage uncertainty in 2 x or '.5 x or π x is the same as that in x. )xample- The following determinations were made in order to find the volume of a piece of wire+ Fiameter+ d = %.11 ( '.'1 mm Nength+ l = J.< ( '.% cm
Aorking in consistent units" and applying the equation V
=
π d
:
l " we have+
V = ::;.J mm 2 The percentage uncertainty" pV = %.< × 1 ) %.' = :.1 > = : > ?to % s.f.@ 4*ote that π and : have no uncertainties.6 So the a!solute uncertainty u = ::;.J × '.': = %I.J5< = 1' ?% s.f.@ So the volume is e#pressed as V = :5' ( 1' mm 2. 7ultiply the percentage uncertainty . Addin$ or sutractin$ 'uantities *A2 only, f 1 quantities are added or su!tracted the a!solute uncertainty is added. This situation does not arise very frequently as most equations involve multiplication and division only. The e.m.f. H p.d. equation for a power supply is an e#ception.
n all cases" when the final > uncertainty is calculated it can then !e converted !ack to an a!solute uncertainty and quoted % sig. figure. The final result and uncertainty should !e quoted to the same num!er of decimal places
3otes for purists%. Ahen working at a high academic level" where many repeat measurements are taken" scientists often use ,standard error- ε , a.k.a. ,standard uncertainty-. Ahere this is used" the e#pression x' ( ε is taken to mean that there is a pro!a!ility that the value of x is in the range x' − ε to x' ) ε " a J5> pro!a!ility that it lies in the range x' − 1ε to x' ) 1ε " a J;> pro!a!ility that it is !etween x' − 2ε and x' ) 2ε " etc. Our work on uncertainties will not involve this highDlevel approach. 1. The method which we use here of estimating the uncertainty in an individual quantity takes no account of the num!er of readings. This is !ecause it is e#pected that only a small num!er of readings will !e taken. Fetailed derivation of standard uncertainties ?see a!ove@ involves taking the standard deviation of the readings and then dividing this !y n − % " so
taking %' readings would involve dividing σ !y 2. 2. The a!ove method of com!ining uncertainties has the merit of simplicity !ut it is unduly pessimistic. f several quantities are com!ined" it is unlikely that the actual error ? sic@ in all of them is in the same direction" i.e. all ) or all −. $ence adding the percentage uncertainties overestimates the likely uncertainty in the com!ination. 7ore advanced work involves adding uncertainties in quadrature+ i.e. p =
p%1
+
p11
+
done when standard uncertainties are employed ?note % a!ove@. t is not intended that candidates pursue any of these coursesR
p2 1
+ ......
. This is normally
G#AP&S *derivation of uncertainties from $raphs is only expected in A2,
The following remarks apply to linear graphs+ The points should !e plotted with error !ars. These should !e centred on the plotted point and have a length equal to yma# − ymin 4for uncertainties in the y values of the points6. f identical results are o!tained the precision of the instrument could !e used. f the error !ars are too small to plot this should !e stated. f calculating a quantity such as gradient or intercept the steepest line and a least steep line should !e drawn which are consistent with the error !ars. t is often convenient to plot the centroid of the points to help this process. This is the point ( x" y ) " the mean x value against the mean y value. The steepest and least steep lines should !oth pass through this point. . The ma#imum and minimum gradients" mma# and mmin" 4or intercepts" cma# and cmin6 can now !e found and the results quoted as+ gradient = intercept =
mma#
+ mmin
1 cma#
+ cmin
1
m ma# − m min 1 c ma# − cmin ± 1 ±
Scales
Graph should cover more than 8 of the graph paper availa!le and awkward scales 4e.g. multiples of 26 should !e avoided. otation of the paper through π H: 4J'° R6 may !e employed to give !etter coverage of the graph paper. Semi4lo$ and lo$4lo$ $raphs *A2 only,
Students will !e e#pected to !e familiar with plotting these graphs as follows+ x SemiDlog+ to investigate relationships of the form+ y = ka . Taking logs+ log y = log k + x log a or ln y = ln k + x ln a 4t doesnBt matter which6 So a plot of log y against x has a gradient log a and an intercept log k . )xamples- adioactive or capacitor decay" oscillation damping n NogDlog+ to investigate relationships of the form+ y = Ax Taking logs+ log y = log A + n log x 4or the equivalent with natural logs6 So a plot of log y against log x has a gradient n and an intercept log A . )xamples- Cantilever depression or oscillation period as a function of overhang length" Gallilean moon periods against or!ital radius to test relationship. *ote that 5o$4lo$ or semi4lo$ $raph paper will not e re'uired. "ncertainties from 5o$ $raphs + Candidates will not !e e#pected to include error !ars in log plots.
Section 2 – 6deas for practical work
Prac Fensity of regular solids 4cu!oids" cylinders6 dentification of material using density.
Conte#t &se of metre rule" callipers" micrometer" !alance nitial work on uncertainties Fensity of liquids and irregular solids &se of measuring cylinders Aeighing a rule !y !alancing a loaded rule &se of P of 7 0cceleration of a trolley on a ramp 4lots of &se of equations of motion graphs to variants here6 determine acceleration Fetermination of g !y simple pendulum *.. *ot on spec !ut a useful intro to oscillation period measurements nvestigation of a compound pendulum or a Fitto pin and pendulum DL characteristics of diodes" lamps etc. &se of ammeters" voltmeters" varia!le resistors" potentiometers 4pots6. dentification of the material of a wire !y Larious ways single measurements H determination of its resistivity against l . &ncertainty com!inations. Lariation of resistance with temperature for a Thermistor not on spec !ut it doesnBt matter metal wire 4copper is good6 andHor thermistor here. Could tie in with potential dividers to design a temperature sensor. Fetermination of resistance of a voltmeter !y R use of a series resistor. nvestigation of currents in series and parallel circuits Fetermination of internal resistance of a Firect use of V = E − Ir or use of power supply % % r = + D use of reciprocals in graphical V E ER work. Sonometer variation of frequency with &se of reciprocals in graphical work length determination of the speed of transverse waves on the metal wire 7easurement of the wavelength of microwaves !y standing waves 7easurement of the wavelength of microwaves !y Fou!le slit ?or NloydBs mirror@ 7easurement of wavelength of a laser !y oungBs slits 7easurement of wavelength of a laser pointer using a diffraction grating 7easurement of refractive inde# of glass or water !y real and apparent depth 7easure refractive inde# of a semicircular glass !lock using ray !o# 4or pinsR6 7easurement of the speed of sound in air using a dou!le !eam CO and two microphones
Section – )xperimental techni'ues
The following is a selection of e#perimental techniques which it is anticipated that candidates will acquire during their 0S and 01 studies. 6t is not exhaustive " !ut is intended to provide some guidance into the e#pectations of the P$2 and P$< e#perimental tasks. %easurin$ instruments The use of the following in the conte#t of individual e#periments+ • micrometers and callipers. These may !e analogue or digital. t is intended that candidates will have e#perience of the use of these instruments with a discrimination of at least '.'% mm. 0 typical use is the determination of the diameter of a wire. • digital topDloading !alances. • measuring cylinders and !urettes. This is largely in the conte#t of volume and density determination. • force meters ?*ewton meters@. • stop watches with a discrimination of '.'% s. t is also convenient to use stopwatches H clocks with a discrimination of % s. • rules with a discrimination of % mm. • digital multimeters with voltage" current and resistance ranges. The following ?d.c.@ ranges and discriminations illustrative the ones which are likely to !e useful+ 1L '.''% L 1' L '.'% L %' 0 '.'% 0 10 '.''% 0 1 k Ω %Ω 1'' Ω '.'% Ω Students should !e familiar with the technique of starting readings on a high range to protect the instrument. • liquid in glass thermometers. D%' − %%'°C will normally suffice" though candidates can !e usefully introduced to the advantages of restricted range thermometers. Ahere appropriate" digital temperature pro!es may !e used.
E#perimental techniques The purpose of P$2 is to test the a!ility of the candidates to make and interpret measurements" with special emphasis on+ • com!ining measurements to determine derived values" eg density or internal resistance • estimating the uncertainty in measured and derived quantities • investigating the relationships !etween varia!les These a!ilities will !e developed !y centres" using all the content of P$% and P$1. They can and will !e assessed using very simple apparatus which can !e made availa!le in multiple quantities. $ence it is not foreseen that apparatus which centres are likely to possess in small num!ers" if at all" will !e specified" e.g. oscilloscopes" data loggers" travelling microscopes. The following list may !e found useful as a checklist. Candidates should !e familiar with the following techniques+ • connecting voltmeters across the p.d. to !e determined" i.e. in parallelM
• • • •
• •
•
connecting ammeters so that the current flows through them" i.e. in seriesM the need to avoid having power supplies in circuits when a resistance meter is !eing employedM taking measurements of diameter at various places along a wire H cylinder and taking pairs of such measurements at right angles to allow for nonDcircular cross sectionsM determining a small distance measurement" e.g. the thickness or diameter of an o!9ect" !y placing a num!er of identical o!9ects in contact and measuring the com!ined value" e.g. measuring the diameter of steel spheres !y placing 5 in line and measuring the e#tent of the 5M the use of potentiometers ?*.. not metre wire potentiometers@ and varia!le resistors in circuits when investigating currentDvoltage characteristicsM the determination of the period and frequency of an oscillating o!9ect !y determining the time taken for a num!er of cycles 4typically %' or 1'6M *.. 0lthough the concept of period is not on the 0S part of the specification" it is likely to !e used in P$2M the use of fiducial marks and noDparalla# in sighting against scales and in period determinations.
