Analysis of Experimental Measurements

Transcription

1 Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example, cosider measurig the legth of a object with a meter stick. You lay the meter stick ext to the object ad determie its legth by how the eds of the object lie up or compare with the marks o the meter stick. The measuremet compares the legth of the object to the legth of a stadard meter. The meter stick is a device that aids i makig the compariso. Because all measuremets are actually comparisos to stadards of physical quatities, all measured values are estimates ad are therefore ot exact values. Sice all measured values are oly estimates ad ot exact values, kowig the estimate by itself is ot sufficiet. It is quite importat to kow how good the estimate is, otherwise, the estimate itself is ot very meaigful. Cosider measurig the height of a doorway. You could make a rough measuremet by usig your body as a crude istrumet if you kow your height. A reasoable estimate of the height of the doorway would be 7 feet. Now alog comes a carpeter that eeds to fit a door ito the doorway. Ca the carpeter use your estimate to cut a door to fit? Your aswer is probably o. You kow your aswer may be off by several iches ad that just would't be close eough for cuttig a door to fit. Uless you idicate to the carpeter how your measuremet was obtaied, the carpeter has o idea how good your value of 7 feet is. Physics, more tha ay other sciece, is based o accurate measuremets of primitive quatities, such as mass, legth, ad time. It is thus appropriate that you gai some uderstadig about the basic priciples of measuremets ad the treatmet of errors or ucertaities that are ivariably associated with these measuremets. First we will discuss sigificat figures. Cosider a measuremet of the positios of the three lies i the figure below. Scale A lie lie lie 5 oe sigificat figure cm 6 cm 9 cm Scale B two sigificat figures. cm 6.5 cm 9.0 cm Scale C three sigificat figures.5 cm 6.50 cm 9.00 cm If we use scale A the we might measure the positios of the three lies as ():, (): 6, (): 9 (i uits of cm), while with scale B we might obtai ():., (): 6.5, (): 9.0. Fially with scale C: ():.5, (): 6.50, (): 9.00.

2 I each case the umbers are the best represetatios of the measuremets with the give scale, limited by the markigs o the scale. Sigificat figures are defied to be the least umber of digits which give the best represetatio of a measuremet i accordace with the accuracy limitatio of the measurig device. I the example above we were limited to oe sigificat figure whe we used scale A while with scale B we had we had two sigificat figures ad with scale C, three sigificat figures. Notice that with scale B, i the case of lie (), or with scale C i the case of lies () ad () we eeded to iclude zeros as sigificat digits i places where ormal usage might lead to their beig dropped. To avoid cofusio as to whether a zero is a sigificat digit a sesible procedure would be to uderlie a termial sigificat zero, e.g. 9.0 or 9.00 for lie () with scales B ad C respectively. This is especially true i the case of umbers like 9000, where ay of the three zeros could be the last sigificat digit; this problem is somewhat alleviated by the use of scietific otatio e.g x 0 would idicate three sigificat figures. I combiig umbers oe must be cogizat of the umber of sigificat figures i each. If we cosider the followig sum i which doubtful digits (lowest order sigificat digits) are uderlied: If you rouded the umbers.574 & 0.6 before addig them, you would get 6.5. By addig ad the roudig, you get a better value 6.4. We see that the last three figures are doubtful. There is o poit i retaiig the last figures i the hudredths or thousadths place sice there is already a doubtful figure i the teths place (the 7 i.7 could be 6 or 8). We thus arrive at the followig rule; Do ot retai ay digits i a additio or subtractio beyod the first decimal place which cotais a doubtful figure. I the above example we thus ca roud all umbers to the earest teth sice the teths place is already doubtful. For multiplicatio or divisio we eed a differet rule. Cosider the product: 4 x (itermediate result) 600 (fial result) We see the ucertai i makes the last three figures ucertai. The product should thus be writte 600. ule: I a multiplicatio or divisio, the umber of figures retaied i the aswer is the umber of sigificat figures i the umber which has the fewest sigificat digits. Thus i the example, the has oly two sigificat figures while 4 has three, the product has oly two. Note that the above rule pertais to fial aswers. If the computatio is ot a fial result but oly a itermediate result it is reasoable to retai oe extra digit i a product or quotiet i.e. the above result could be rouded to 60 if it were to be used further.

3 Measurig ucertaities ad errors. The result of a measuremet yields a umerical represetatio i which the umber of sigificat digits is usually determied by the measurig istrumets ad our ability to read them. The last sigificat digit o the right (the least sigificat digit) has a ucertaity or error associated with it which ca be illustrated by the example of the precedig paragraph. Whe scale A was used to measure the positio of lie (), the value was ascertaied by the experimeter to represet his judgmet of the readig o his scale, but the actual value could be.5 or or. or 4, although values less tha or more tha 4 would seem ulikely. Thus the error i the value uits is about cm either directio, so we write ± to idicate our ucertaity. Similarly i the uses of scales B ad C with the same lie we might write. ±. ad.5 ±.0 respectively, depedig o how accurately we believe we ca read the scale. The errors just discussed are ofte overshadowed by other errors, caused by a collectio of ucotrolled factors such as vibratios of istrumets, air currets, temperature fluctuatios, etc. For example we might eed to measure the legth of a object which is loger tha our ruler, or a legth alog a curved surface. To accomplish this we might use a piece of strig stretched betwee the poits desired, ad the use the ruler to measure the strig. Whe we do this we have o assurace that the strig was put uder the same tesio each time. We thus are led to repeatig the measuremet several times. The fact that we get slightly differet results each time idicates that the ucotrolled factors are preset. I this case statistics idicates that the best estimate of the true legth or distace is ofte the average or arithmetic mea of the results of the various trials: Average x x i i, () where the x i are the idividual measuremets ad is the total umber of measuremets. (Each x will be stated to a umber of sigificat figures give by the accuracy limit of the apparatus. The average x should be rouded to the same accuracy.) The experimetal values will differ from the average by a amout ( x i - x ) called the deviatio of the measuremet from x i. A simple measure of the error i the measuremets is give by the average deviatio defied as the average of x i - x (i. e. the deviatios, all take as positive): Average deviatio, a.d. x i x. () Note: The average deviatio is actually a rather crude estimate of the error i the measuremets. Accordig to statistical theory a better measure is give by the stadard deviatio, defied by: Stadard deviatio σ x i i ( x i x). () However sice σ x is harder to compute tha the average deviatio (a.d.), ad usually does ot differ from it by more tha about 5%, i this laboratory we will be satisfied to use the average deviatios. Some pocket calculators have the σ x fuctio built i. If you have access to such a calculator you may quote the σ x i place of the average deviatio i all labs except this lab. While the average deviatio (a.d.) gives a estimate of the accuracy of idividual measuremets, we still have ot assiged a error to x. If we take a large umber of measuremets we would expect x to be

4 very close to the true value, while the average deviatio should ot be very differet from the result with oly a few measuremets. Statistical theory tells us that the best error to assig to the mea x is the average deviatio of the mea (a.d.m.) defied by a. d. m. a. d., (4) where is the total umber of measuremets. Thus as icreases, while the a.d. does ot vary much, the a.d.m. decreases ad the faith we would place i x becomes greater. However, the ultimate accuracy i x is still limited, of course, by the umber of sigificat figures i the measuremets, so there is o poit i lettig get to be too large. As a example, suppose we make four measuremets of the legth of a object, with the results below (i meters): measured values (x i ) deviatios ( x i - x ) )58.0 ( ) a.d 4._0 4.0_ x 4.55 m a.d.m..0_ / 4.5 Note that we have kept oe extra figure beyod the sigificat figures i x i whe computig x, the deviatios, the a.d. ad the a.d.m. But it would ot be proper as a fial result to express the measured legth as 4.55 ±.5 m because there are ot this may sigificat figures possible. The result should be rouded to 4.6 ±. m. If may more measuremets were made the a.d. would remai.0_ or thereabouts, while the a.d.m. would decrease. But after 9 measuremets the a.d.m. would be less tha. ad further improvemets would ot be realistic because this is the limit of the istrumet as idicated by the sigificat figures. Note also that i computig the a.d. we took the absolute value of all the deviatios i.e. treated them as if positive. It should be oted that these techiques are oly valid if the experimet ca be repeated a umber of times ad if the errors are truly radom. Not icluded i this aalysis are systematic error i.e. ukow ad ucotrolled biases, which shift all the measuremets off i the same directio. A example would be a woode ruler, which was improperly calibrated or had shruk. Nor are we icludig bluders e.g. readig the rule improperly (e.g..6 for 4.6 m). Systematic errors are particularly difficult to detect ad elimiate, ad may seemigly fie experimets have proved to be erroeous because of some ususpected systematic bias i the procedure.

5 Error Propagatio i Calculatios Suppose that the legth measured i the precedig paragraph, a 4.6 ±. [m], is oe side of a rectagle, the other side of which is b 6.4 ±.0 [m]. From these two umbers we ca compute the perimeter of the rectagle P (a + b) or its area A ab. The questio arises: how do we compute P ad A with our ucertai values of a ad b, ad what errors do we assig to the computed values? Let us look first at P. If we use the mea values for a ad b we fid P ( ) 4.88 [m]. Statistical theory ideed says that this is the best value for P, provided that the errors i a ad b are ot correlated. (As we saw earlier, the 8 i the hudredths place is ot sigificat sice a was accurate oly to the teths place, so we should roud this value to 4.9 [m].) With regard to the error to be assiged to this value, we could take the maximum values a , b ad yield P 4. [m] ad take miimum values a , b ad yield P 4.44 [m]. The error is thus ±.44 [m] ad P 4.88 ±.44 [m], or roudig, P 4.9 ±.4 [m]. The ucertaity i P is thus ( a + b) where a ad b are the ucertaities i a ad b. We are thus lead to the simple rule: The error assiged to a + b is a + b. Ad the error assiged to a (where is a exact quatity like the i the formula for P) is a. Try to covice yourself that the error assiged to - a is a (that is why we have placed the absolute value sig o ) ad thus that the error assiged to a - b is a + b. We always add the errors, ever subtract them. Note: statistical theory shows that this estimate of the error i a ± b is too crude sice it is based upo both values of a ad b beig at their most extreme possible values, which is ulikely. The error a + b is likely to be a overestimate, ad a more reasoable value for the error i a ± b is ( b a) + ( ). (5) Now cosider the area A ab. Agai statistical theory says that the best value to be assiged to A is the product of the mea values a b. What error shall we assig to this? Takig our example A (4.6)(6.4) [m ]. As we saw earlier, i a product the umber of sigificat figures is the same as that i the factor with the least umber of sigificat figures. I our case both factors have three sigificat figures so we should roud A to the same accuracy: A 9.6 [m ]. Istead of calculatig the error i A umerically let us do it symbolically: ( a ± a)( b ± b) ab ± a b ± b a ± a b. Geerally the last term is so small that it may be dropped. We thus see that the possible rage of values of A is from ab ( a b + b a) to ab + ( a b + b a). We thus assig to A the error A ( a b + b a). There is a better way of expressig this: use the fractioal or relative error. If we divide A by the assiged value of A ( ab) we get A a b +. (6) ab a b The fractioal error i A is the sum of the fractioal errors i a ad b. I our umerical example the fractioal error i a is./ (or.4 %), ad i b is.0/ (or.%). The fractioal error i A ab is thus or.7%. The absolute error i A is thus A (.07)(9.6).6 [m ], roudig off to oe decimal place. Thus we get A 9.6 ±.6 [m ] or 9 ± [m ]. By argumets such as these we arrive at certai rules, which we summarize i the ext page:

6 ule : Whe the result of a calculatio ivolves the additio or subtractio of iexact umbers, the ucertaity i the result is the sum of the ucertaities of the idividual terms. If A ± B ± C... the ( A) + ( B) + ( C) +... A + B + C +... (7) emember that the last expressio is just a approximatio. ule : Whe the result ivolves the product or quotiet of iexact umbers, the fractioal ucertaity i is calculated as the followig: if AB, C A A B + B C + C A B C + +. (8) A B C (The rule for the ucertaity i a, where is exact, follows from ule with equal to zero.) ule : Whe ivolves powers, the fractioal ucertaity i is calculated as below: if p q A B A B C A B C, the p q r p + q + r. r + + C A B C A B C (9) ule 4: ule 5: ule 6: Whe ivolves trig fuctios of a iexact argumet we use the followig rules ( A i rad): If si A, the cos A A. (0) If cos A, the si A A. () Errors or ucertaities ( A, etc.) are always treated as positive. Always roud off fial results to the appropriate umber of sigificat figures. We coclude with two comprehesive examples. Example : Suppose (A + BC)/D Let N / ( + )/, where N +, A, BC, ad D. First of all, A ad D. Ad usig rule, The, Example : Suppose Let N N + (A si B) C BC, where ( ) ( +. A C ad si B, so. The by ule, A C By ules ad, we fid +, A C cos B B while by ule 4, cos B B, or cot B B. si B N ) B B C +. C +. Note that the factor of does ot eter the formula for the fractioal error. Whe we multiply through by to get the absolute error i,, the will eter properly.