Activity 3: Length Measurements with the Four-Sided Meter Stick

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1 Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter stick are arithmetically maipulated. THEORY: Error of measuremet refers to the ucertaity of a readig. It carries o implicatio of mistake or bluder o the part of the observer. Error meas the ucertaity betwee the measured value ad the stadard value. It ca either be a positive error, that is the error that teds to make a readig too high, or a egative error, oe i which readig teds to be too low. Errors may be grouped ito two classes, systematic ad radom. A systematic error is oe that always produces a error of the same sig, for example, oe that would ted to make all the observatios too small. A radom error, also called erratic error, is oe i which positive ad egative errors are equally probable. Systematic errors ca be subdivided ito three groups depedig o the sources of errors. These are: 1. Istrumetal Errors This type of errors is caused by faulty or iaccurate apparatus. For example, a udetected zero error i a scale, a icorrectly adjusted or a meter with udue frictio. 2. Persoal Errors These errors are due to some peculiarity or bias of the observer. Probably the most commo source of persoal error is the tedecy to assume that the first readig take is correct ad to look with suspicio o ay variatio from this readig. Other persoal errors may be due to fatigue, eyestrai, or the positio of the eye relative to a scale. 3. Exteral Errors Errors of this type are caused by exteral coditios such as wid, temperature, humidity, vibratio, etc. Examples of exteral errors are the expasio of scales as the temperature rises ad the swellig of meterstick as humidity icreases. Correctios may be made for systematic errors whe they are kow to be preset. Radom or erratic errors occur as variatios that are due to a large umber of factors, each of which adds its ow cotributio to the total error. Iasmuch as these factors are ukow ad variables, it is assumed that the resultig error is a matter of chace ad therefore positive ad egative errors are equally probable. Because radom errors are subject to the laws of chace, takig a large umber of observatios may lesse their effect i the experimet.

2 Due to the fact that variatios i the observatios oly follow chace, the laws of statistics may be used to arrive at certai defiite coclusios about the magitude of the errors. The followig results of statistical laws ca be utilized to quatify error of measuremet: 1. Arithmetic Mea, a.m. The arithmetic mea a.m. represets the best value obtaiable from a series of observatios. It is the value havig the highest probability of beig correct. It is obtaied by dividig the sum of the idividual readigs by the total umber of observatios. I symbol, 2. Average Deviatio, a.d. N a.m. The average deviatio a.d. is a measure of the accuracy of the observatio. It shows the average value of the divergece of the observatios from the arithmetic mea. The average deviatio is calculated by dividig the sum of the deviatios (without regard to sig) by the umber of observatios, where deviatio d refers to the differece betwee a observatio ad the arithmetic mea. I symbol, 3. Average Deviatio of the Mea, A.D.. d a.d.. The average deviatio of the mea A.D. measures the heterogeeity or eveess withi a set of observatios. If the average deviatio of the mea is small, the it ca be said that the set of observatios is homogeeous. For it is kow from theory of probability that a arithmetic mea computed from equally reliable observatios is o the average more accurate tha ay oe observatio by a factor of, the average deviatio A.D. of the mea of observatios is give by a.d. A.D.. The importace of a error i a experimetal value is ot i its absolute value but i its relative value. By percetage error is meat the umber of parts out of each 100 parts that a umber is i error. The equatio for solvig the percetage error is Percetage Error = Experimetal Value Stadard or Accepted Value Stadard or Accepted Value x 100 Whe two values of the same quatity are obtaied experimetally, the percetage differece is used to compare the two values. The equatio is Percetage Differece = Differece betwee the two values Average of the two values x 100

3 I additio, whe experimetal data are arithmetically maipulated, errors may be propagated. The followig rules are safe to apply i makig computatios: 1. I additio ad subtractio, carry the result oly through the first colum that cotais a doubtful figure. 2. I multiplicatio ad divisio, carry the result to the same umber of sigificat figures that are i the factor with the least umber of sigificat figures. Oe must lear therefore how to determie sigificat figures i a measuremet. The sigificat figures of a umber specify its accuracy, the umber of digits which are meaigful. For example, if you have a ruler marked off i millimeters ad you use it to measure the legth of a lie, you are able to measure accurately oly to the earest millimeter. Suppose you foud the lie to be 1351 mm or m log. The umber of sigificat figures i this case is 4 (1, 3, 5 ad 1). It would ot be correct to write mm or m, sice this implies that you kow the legth to a greater accuracy tha you actually do. The trickiest aspect of sigificat figures is kowig how to treat zero. For example, the umber of sigificat figures i the umber 7300 is two, while the umber of sigificat figures i the umber 7301 is four. This is because the zeros i 7300 are simply place holders, givig the magitude or power of 10 for the umber. Thus 7300 implies that this umber is kow oly to the earest power of 2 (100's). O the other had, 7301 implies that the umber is kow to the earest uit. The same situatio holds for umbers less tha oe. For example, the umber has three sigificat digits, while the umber has oly oe. The zeros to the right of the decimal do ot specify accuracy but, rather, are place holders, specifyig the magitude (powers of 10) of the umber. MATERIALS: The materials eeded for the experimet are: woode block four-sided meter stick PROCEDURE: 1. Use the four-sided meter stick to measure the block, makig the same measuremet successively by meas of scales of progressively fier graduatios. First, measure the legth usig the face with o graduatio. This measuremet is oly a estimate of a fractio of a meter. Record this legth ad the three followig sets of data i rows as follows: (a) observed legth, icludig the portio of the smallest scale divisio; (b) the value of the smallest scale divisio beig read; (c) the fractioal part of the scale divisio that ca be by the eye with reasoable certaity (for example, 0.1 of a scale divisio); (d) the umerical ucertaity of the observatio, which will simply be (c) expressed as a actual legth, for example 0.1 m; (e) the percetage ucertaity ivolved i the estimatio, which will be obtaied by dividig (d) by (a) ad multiplyig by 100 per cet. 2. Repeat the measuremet of the legth of the block, this time usig the face of the stick calibrated i decimeters. Estimate the ucertaity, ad record all the data, exactly as i the previous step. Repeat, usig the face graduated i cetimeters. Repeat, usig the millimeter scale.

4 3. Measure ad record the legth of the block usig the millimeter scale. Take six observatios at differet places o the block i order that a fair average may be obtaied. I each readig estimate fractioal parts of millimeter divisios. (Remember to iclude the proper umber of zeros whe the observed legth seems to fall exactly o a scale divisio.) Calculate the mea of the observatios of the legth. Fid the deviatio from the mea ad calculate the average of the deviatios. Compute the A.D. of the mea legth. Attach this A.D. to the mea legth with a ( ) sig ad call this fial result the observed legth of the block. 4. Repeat Procedure 3 for the width ad thickess of the block ad determie the A.D. of each. Calculate the volume of the block, recordig the result to the proper umber of sigificat figures. Determie the umerical ad percetage errors of this volume, usig for the respective errors of the legth, width, ad thickess the A.D. s foud above. DATA Table 1. Measuremet of Legth Usig a Meterstick with No Graduatio Table 2. Measuremet of Legth Usig a Meterstick Calibrated i Decimeters Table 3. Measuremet of Legth Usig a Meterstick Calibrated i Cetimeters

5 Table 4. Measuremet of Legth Usig a Meterstick Calibrated i Millimeters Table 5. Measuremet of Legth of a Block Trials Observed Legth Deviatio from the Mea Average a.m. = a.d. = Average Deviatio of the Mea A.D. = Table 6. Measuremet of Width of a Block Trials Observed Legth Deviatio from the Mea Average a.m. = a.d. = Average Deviatio of the Mea A.D. = Table 7. Measuremet of Thickess of a Block Trials Observed Legth Deviatio from the Mea Average a.m. = a.d. = Average Deviatio of the Mea A.D. =

6 Table 8. Propagatio of errors durig arithmetic maipulatios Volume (accepted value) = mm 3 Volume Miimum Maximum Calculated Value Numerical Error Percetage Error GUIDE QUESTIONS FOR ANALYSIS: 1. How does the side of the four-sided meter stick ifluece the precisio of your measuremet? How about the accuracy? 2. Five differet studets take the followig measuremets of the same object: 1.0 m 1.45 m 1.5 m m 1.46 m (A) Why are the measuremets differet? (B) Which oe is correct? (trick questio) 3. Discuss the relative accuracy obtaied ad the variatio i the error caused by estimatig fractioal parts of scale divisios as the same legth is measured with scales of progressively decreasig legths of scale divisios (refer to the data gathered i Procedure 2 ad Procedure 3). 4. What does the average of the deviatios sigify? 5. What is the sigificace of average deviatio of the mea?

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