THE NEED FOR MEASUREMENT

AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS THE NEED FOR MEASUREMENT The heart and sole of any scence s measurement and expermentaton. In scence, attempts are made to make sense of our physcal world by usng theores that are support by expermental data. We are able to examne our envronment by usng our senses. Unfortunately our senses are easly msled, they do not always provde relable nformaton. For example, f you carry a heavy sutcase for a long perod of tme and then pck up book, t wll feel lght; eatng a pece of cake before and after cleanng our teeth wth toothpaste wll gve dfferent taste sensatons; prolonged drvng n a car at 110 km.h -1 can alter one s percepton of speed when you have slowed down to 50 km.h -1 ; judgment of temperature by touch may depend upon whether you have had your hand n cold or hot water. AUSTRALIAN CURRICULUM PHYSICS 1

Look carefully at the followng fgures. AUSTRALIAN CURRICULUM PHYSICS 2

Clearly, observatons have to be nterpreted wth cauton. If nformaton s to be useful t must be objectve and reproducble; t must be free from the whms and peculartes of the observer. Observatons must develop nto somethng more quanttatve t must become measurement. AUSTRALIAN CURRICULUM PHYSICS 3

RELIABILITY OF MEASUREMENT There are three purposes n consderng the relablty of measurements: 1 Comparson wth other results. 2 Comparson of a measured quantty wth a theoretcal predcton. 3 A pece of expermental data may be requred for the desgn of some apparatus or procedure e.g. an engneer needs to know how much confdence he can place n the values of the strength of materals n the constructon of a brdge. A measurement s the result of some process of observaton or experment. The am of any measurement s to estmate the true value of some quantty. However, we can never know the true value and so there s always some uncertanty that s assocated wth measurement (except for smple countng processes). A rough method of ndcatng the degree of uncertanty n a measurement s through the number of sgnfcant fgures. The usual conventon s to quote no more than one uncertan fgure. For example, g = 9.81 m.s -2 the dgt 1 s n doubt and g s expressed to 3 sgnfcant fgures. A better method of ndcatng uncertanty n a measurement s to quote t explctly e.g. the length of the box s (10.00 0.05) mm. Ths means that we are reasonably confdent that the length of the box s between 9.95 to 10.05 mm. N.B. the value for the measurement and ts uncertanty are quoted to the same not of decmal places and not the same number of sgnfcant fgures. As a general rule only or possbly two sgnfcant fgures are gven for the uncertanty. A scentfc value (physcal quantty) thus needs four Parts: name and/or symbol value uncertanty unt There are many sources of uncertanty (errors) n measurements. Often, we consder only two categores: random errors and systematc errors. Random errors arse through repeated measurements of the same physcal quantty. The random error gves the precson of a measurement. the smaller the random error then the better the precson Precson s the reproducblty of a measurement under dentcal crcumstances. It has nothng to do wth correctness, an ncorrect result can be hghly reproducble. AUSTRALIAN CURRICULUM PHYSICS 4

Systematc errors are nherent n the system and can t be removed or even detected through repeated measurements. The systematc error gves the accuracy of the measurement. the smaller the systematc error then the better the accuracy Accuracy refers to the correctness of a measurement or absence of bas. Systematc errors shft the randomly spread readngs away from the true value and so no matter how many readngs are taken, the fnal result wll not approach the true value. Random errors are dealt wth statstcally whereas systematc errors are trcky and they depend largely on the equpment used and on the skll and technques of the expermenter. Thus, the uncertanty n a measurement depends upon both the random errors and systematc errors. accurate & precse measurement mean not accurate & precse measurement mean `true vale `true vale accurate & not precse measurement mean not accurate & not precse measurement mean `true vale `true vale AUSTRALIAN CURRICULUM PHYSICS 5

Examples of systematc errors Human Error Many errors arse n the actual process of human observaton such as reacton tme n usng a stopwatch or a parallax error n readng an analogue meter. Repeated readngs of the same quantty often reveal ths type of mstake. It s more dffcult to detect an error n technque whch leads to all the repeated readngs beng based n the same way. correct readng: eye must be placed drectly above ponter on the meter wrong readng: gves rse to parallax error Instrument Lmtatons Instruments vary markedly n qualty, some are desgned better and ncorporate better workmanshp and components. The accuracy of a mcrometer depends on the unformty of the screw thread. Many electrcal meters used n schools have are accurate to more than 3%. For example, a voltmeter has an accuracy of 3% and a voltage of 10.0 V s recorded. Ths mples that the true readng s not 10.0 V but les n the range 9.7 V to 10.3 V. AUSTRALIAN CURRICULUM PHYSICS 6

Not measurng what you thought you measured Current flows from the batter through the ammeter A and then splts, some gong through the voltmeter V and some gong through the resstor R, before rejonng and flowng back to the battery. The ammeter readng A s often used to measure the current through the resstance R. A V R Zero errors An nstrument may not be zeroed, and therefore you have to make a correcton for ths. You should not make correctons for zero errors n your head. Always record the actual nstrument readng, and apply the correcton separately. Calbraton Errors It s often necessary to calbrate an nstrument. Ths can be by careful comparson of the nstrument wth those obtaned on an nstrument known to be accurate. Example A ammeter was used to measure the current through a resstor. Recorded results: Zero readng: -0.03 A Calbraton factor: 0.90 (lnear) Instrument readng: 1.06 A Corrected readng: (1.06 + 0.03)(0.9) A = 0.98 A Extraneous nfluences Often an experment goes wrong because the quantty beng measured s nfluenced by extraneous nfluences. For example an electronc top pan balance s senstve to movement n the room; expanson of a metal ruler due to an ncrease n temperature; mpurtes present when dong a chemcal test. AUSTRALIAN CURRICULUM PHYSICS 7

Dsturbances caused by the act of observaton The very act of makng an observaton or measurement can alter the stuaton beng studed. for example the study of anmals under controlled condtons; the measurement of tyre pressure wth a gauge; mmerson of a cold thermometer n a beaker of hot lqud; mcrometer squashes paper when measurng the thckness. No observaton can be made wthout alterng what s beng observed. Even when an object s smply llumnated n order to look at t - the lght exerts a force on t and alters the state of the object beng measured. In the theory of Newtonan mechancs t s assumed that one can measure the poston and momentum of a partcle exactly. However, ths assumpton fals when appled to atomc systems. The better the poston of a partcle s known then the greater the uncertanty n the momentum of the partcle. Ths dea s one form of the Hesenberg Uncertanty Prncple. RANDOM ERRORS AND PRECISION The resoluton s a statement about the precson of a measurement. The resoluton depends on: the graduaton of the nstrument the physcal quantty to be measured the care and skll of the observer. In general lmts of resoluton are sgnfcant only when only one readng has been taken or when repeated readngs are all dentcal. The resoluton should not be taken as half the smallest scale dvson on an nstrument as s often stated. Instead you should quote the amount by whch the readng would have to change before you could notce that change and thong about how many dgts are sgnfcant n the measurement. For example, you want to measure the wdth of an A4 page wth a rule graduated n mllmetres. The resoluton s not 0.5 mm (half the smallest scale dvson) but about 1 mm because of the dffculty of algnng the ruler wth the page. The process of duplcatng measurements s at the heart of any expermental process. Usually when you take a set of measurements there are fluctuatons n the results,.e., not all recordng are the same. If the fluctuatons n the measurements are small then the precson sad to be good. AUSTRALIAN CURRICULUM PHYSICS 8

We usually assume that the varaton n the measurements s descrbed by a normal dstrbuton. For large data sets the normal dstrbuton s shown graphcally as a bell shape curve wth the x-axs correspondng to the measurements and the y-axs to the frequency of each measurement. The mean corresponds to the peak n the curve because most measurements are centred about the mean value. The curve s symmetrcal about the mean and very few measurements are far from the mean. Examples of normal dstrbuton nclude the heghts of all adults n Australa and the marks of students dong ther fnal hgh school examnaton. The best estmate of the measurement s usually taken as the mean (average) value. If the set of measurements are x 1, x 2, x 3, wth occur wth frequences f1, f2, f3, then the mean value x s defned as 1 n x fx where x s the th readng and the total number of readng s n f The precson depends upon the spread of the measurements about the mean value. The spread can be measured by takng the maxmum devaton of all measurements from the mean. Ths quantty s s called the standard devaton and s gven by the equaton s 2 f x x n AUSTRALIAN CURRICULUM PHYSICS 9

frequency The standard devaton s s taken as the characterstc wdth of the normal dstrbuton. A large value of s corresponds to a wde curve and a small value of s the curve s narrow. NORMAL DISTRIBUTIONS small standard devaton large standard devaton measurements AUSTRALIAN CURRICULUM PHYSICS 10

frequency Assumng that the measurements are descrbed by a normal dstrbuton (large number of readngs) then you can conclude that 68% of the ndvdual readngs are n the range x s 95% of the ndvdual readngs are n the range x 2s 99.7% of the ndvdual readngs are n the range x 3s mean x x-2s x-s x x+s x+2s measurements x There are a number of ways whch are scentfcally acceptable n recordng a measurement for random errors. Quotng the correct number of sgnfcant fgures e.g. mean voltage V = 10.5 V (3 sgnfcant fgures) Quotng the range of measurements from the extreme values e.g. mean voltage V = (10.5 0.8) V Quotng the standard devaton e.g. mean voltage V (10.5 0.2) V AUSTRALIAN CURRICULUM PHYSICS 11

EXAMPLE Image that you perform an experment to measure the acceleraton due to gravty g by tmng an object to fall 2.00 m. You know the acceptable value for to be g =9.81 m.s -2. Below are ways n whch students mght have recorded ther measurement of g : Student 1 g = 9.6474341 ncorrect no unts and too many sgnfcant fgures Student 2 g = 9.6 m.s-2 correct - unts and approprate number of s. f. the dgt 6 s n doubt In the above cases the measurements recorded do not agree wth the acceptable value. Therefore, t s dffcult to come to a scentfc concluson n comparng the recordng wth the acceptable value. Student 3 entered the data n MS EXCEL spreadsheet A good method to analyse expermental data s to use a spreadsheet. You can use the spreadsheet to perform many calculatons such as fndng the mean (average) and the standard devaton as shown n below. From the spreadsheet t can be seen that: mean = 9.8 m.s -2 standard devaton = 0.4 m.s -2 max devaton from mean = 0.6 m.s -2 The result can be recorded as g = (9.8 0.6) m.s -2 g = (9.8 0.4) m.s -2 uncertanty = max devaton from mean uncertanty = standard devaton These two ways of wrtng the measurement of g are much better snce they clearly ndcate the range of values for the measurement. We can clearly state that wthn the random errors assocated wth our experment the results obtan are n agreement wth the accepted value. Quotng the standard devaton as the uncertanty s better than the maxmum devaton snce about 68% of the ndvdual recordng are expected to be n the range from x s to x s. AUSTRALIAN CURRICULUM PHYSICS 12

Sample spreadsheet row/column B C 3 # g (m.s -2 ) 4 1 9.6 5 2 9.4 6 3 9.9 each measurement 7 4 9.5 to 1 decmal place 8 5 9.8 9 6 9.7 10 7 10.4 11 8 10.1 12 9 9.4 13 10 9.6 14 mean 9.7 =average(c4:c13) 15 st dev 0.3 =std(c4:c13) numbers formatted so uncertanty 1 sgnfcant fgure AUSTRALIAN CURRICULUM PHYSICS 13