Physics 215 - Experiment 1 Measurement, Random Error .

Physics 215 - Experiment 1Measurement, Random Error & Error analys isAdvanced reading- from Physics by Giancoli, 6th Edition(Sections 1-4, 1-5 & 1-6)Part A-Measurement of Length and Error AnalysisEquipment:1 Ruler1 Vernier Caliper1 Micrometer CaliperSeveral Coins.Objective:The object of this experiment istwofold:1. To learn to measure lengthsusing a ruler, vernier caliper,and micrometer caliper.2. To become acquainted withtypes of error and statisticalmethods for analyzing one'sdata and for estimating its accuracy.3. To determine the density of ablock of metal.Theory:In using a ruler three thingsmust be remembered: (1) thereading should be estimated toone half of the smallest division;(2) the ends of the ruler shouldnot be used since the ends mayhave become damaged and nolonger be square; (3) errors ofparallax should be avoided byplacing the scale against the object to be measured.In using a vernier calipertenths of a division are not estimated; they are read off thevernier scale. Notice that 10 divisions on the vernier scale corresponds to 9 divisions on the mainscale. Therefore, the mark on thevernier scale which best lines upwith a mark on the main scalegives the reading of a tenth ofthe smallest division on the mainscale (see fig. 1. 1).Figure 1-1In using a micrometer caliper,centimeters and tenths of a centimeter are read from the scale onthe barrel. Then thousandths ofa centimeter are read from thescale on the thimble. Since thisscale only goes from 0 to 50 thousandths the thimble must beturned twice to move one-tenth of

Physics 215 - Experiment 1Measurement, Random Error & Error analys isa centimeter. If the scale is overhalfway between the marks onthe barrel, then 50 thousandthsmust be added to the reading.Ten-thousandths of a centimetershould be estimated. (See fig.1.2.) A zero correction for the micrometer caliper should be determined and recorded. For example, if the micrometer caliperreads 0.002 cm when closed, thenevery reading will be too large bythis amount and the zero correction must be subtracted fromeach reading. When closing themicrometer caliper the smallknurled knob must be used sothat the caliper will not be damaged by overtightening.asx 1 n1x i (x1 x 2 . x n )!n i 1nx will be the most probable valuefor the quantity being measured.By itself, however x gives no indication of the reliability of theresults, that is, of what statisticalerror there may be in the results.To analyze this facet of the problem one needs the standard deviation or root mean square ofthe data.Figure 1-2Statistical Analysis Of Dataand ErrorsMeanIf one makes a series of nmeasurements with results xl,x2,.x n, the mean, or average value,x , of the measurements is definedStandard Deviation Or RootMean SquareThe standard deviation (orroot mean square) of the above nmeasurements is defined as1 (x " x)2 ' 2# 1)! &&% ( n " 1) )(

Physics 215 - Experiment 1Measurement, Random Error & Error analys isσ is a measure of the scatter to beexpected in the measurements. Ifone measured a large number ofvalues xl, then statistically about67% of the xl's would lie betweenx - σ and x σ and about 97% ofthe x i 's would lie between x - 2σand x 2σ.The error discussed here is experimental, or random errorwhich results because one doesnot always get the same result inmaking a series of measurements.This type of error is unavoidablebecause, no matter how accurately one makes one's measurements, there will always be someuncertainty in the measurements.The above error is not the onlytype of error which may be present, however. Systematic errorsmay be present, and if they arepresent they may be difficult toaccount for unless one is aware ofthem. A few examples will explain what systematic error is.The speedometer in most automobiles reads too high so thatone's speed is systematicallylower than the indicated speed,and the speedometer introduces asystematic error into any calculations of the speed of the auto. Asecond example is using a steeltape measure in cold weather; thetape must have contracted due tothermal effects so that its lengthis shorter than the length indicated on the rule. A final example is presented by most electricalmeters which have an indicatedaccuracy stamped on the meter.If the accuracy is indicated as 5%then one's measurements madewith the meter may be too largeor too small by up to 5%. Oneway of compensating for systematic errors is to calibrate one's instruments if more accurate results are desired.Personal or human error is athird type of error. Often theperson taking data is biased bythe first result obtained. In taking measurements one should nottry to make them all come out thesame, but should merely makeeach measurement as accuratelyas possible. Another type of human error is to be sloppy in one'sexperimental technique; for example, if one allows parallax errors to occur in making a measurement, this is an avoidablehuman error. In the present experiment, one must be careful toavoid parallax errors.In analyzing the error in yourdata try to estimate the magnitudes of the various kinds of errorwhich may be present and discuss

Physics 215 - Experiment 1Measurement, Random Error & Error analys isthem in your laboratory report.Procedure:and I.B using the micrometercaliper this time. Record the results on your data sheet.A.RulerMeasure the diameter of a cointhree times with the ruler recording the results on your datasheet, then let your partner dothe same recording his results.Make your measurements at different points so that a good average dimension will be obtained.You should always be able to estimate the fractional part of thesmallest division to get your lastsignificant figure.Calculate the mean diameterand the standard deviation. Discard any nonsignificant figuresbefore recording the mean diameter.D. Measurement of DensityDensity is defined as the ratioof the mass of an object divided by its volume. Using thetriple beam balance, determine the mass of your coin,and then assume it is a cylinder and determine its volume.Try to identify the compositionof your coin from the densityyou calculated.Note:Coins are made of various alloys. However, you should beable to determine the mostabundant metal used in theminting of the coin. Youshould be able to find the exact composition on the web.B.Questions:Vernier CaliperMake three measurements ofthe diameter of the coin as in partI.A but using the vernier caliperthis time. Let your partner dothe same. Record the results onyour data sheet. Calculate thestandard deviation and mean diameter as in part A.C.Micrometer CaliperRepeat the measurements andcalculations made in parts I.A1. Which of the three instruments used in today’s lab do youthink allowed you to make themost accurate measurement ofthe diameter of the coin? Why?2. List and discuss the differenttypes of errors that were presentin your measurements

Physics 215 - Experiment 1Measurement, Random Error & Error analys isPart B- Random Error AnalysisObjectiveThe purpose of this experimentis to make a series of measurements involving a sufficientnumber of trials to permit the useof a statistical theory of errors toevaluate the results.vertical column. We will now obtain two numbers which will givea measure of the variability ofyour skill in this experiment.Part 1:Equipment:Steel BallCarbon PaperSheets of Ruled PaperrulerProcedure:Place a sheet of paper over alayer of carbon paper approximately 30 cm from the table onthe laboratory floor. Mark a lineon the paper which is parallel tothe edge of the table. Using aplumb bob, locate the position ofthe edge of the table on the floorand accurately measure x, thedistance from the table and attempt to hit the line on the paperas the ball strikes the floor.Measure the horizontal distancefrom the position of the edge ofthe table to the actual impactpoint, call it x1 (measure to thenearest cm). Repeat these in axBall shown on table with paper beneath.1) Calculation of the averagevalue:Nx !xi 1Ni

Physics 215 - Experiment 1Measurement, Random Error & Error analys isThe xi’s are simply the measured values for x for the differenttrials. A comparison of this valuewith the true distance from thetable edge to the line showswhether or not the results areconsistently too short or too long.Include on the graph:1) The average value of themeasured value of x ( the arithmetic average).2) The true value of x (actualposition of the line).2) Calculation of the standarddeviation: This quantity gives anindication of the consistency ofthe trials.3) The calculated standard deviation.Write your result as:Part 2N# ( x " x)2i!2 i 1N "13) Plot of the distribution of hitsversus position (i.e., a histogram):Draw a graph of the numberof times the ball hit within aspecified distance from the lineversus the distance from the tablein centimeter intervals.Equipment:Compressed PillsDigital Balance (0.001g resolution)Procedure:Measure the mass of at least30 of these compressed pills andcalculate:a) the average value of themassb) the standard deviation ofthis distributionc) plot the distribution of massversus number for the pills inyour measurement set.Questions:1. Compare the graph of yourdata with the sample graph. Ex-

Physics 215 - Experiment 1Measurement, Random Error & Error analys isplain the differences in the distributions observed. How could youreduce the value of σ if the experiment were repeated?2. If a die were tossed twice,what can you say about the average value of the numberthrown? If the die were tossed100 times, what would be theaverage value of the numberthrown? Why are your answersdifferent?3. What can you say about thedose delivered by a pill in yourmeasurement set. How does thisexperiment help to describe thevariability or consistency of theproduction process producing thismedication?

Physics 215 - Experiment 1 Measurement, Random Error & Error analysis σ is a measure of the scatter to be expected in the measurements. If one measured a large number of