ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision  A measure of agreement between repeated measurements (repeatability).


 Lillian Gibbs
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1 INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie the diameter ad mass of a certai product that is massproduced to isure these products meet the iteded specificatios. The tools used to measure legth will iclude a liear scale ad micrometer. DEFINITIONS Precisio  A measure of agreemet betwee repeated measuremets (repeatability). Error  The differece betwee the true value ad the measured value. Accuracy  Agreemet betwee measured ad true values (the absece of error). Resolutio  The smallest step or iterval that we ca measure or distiguish o the object or parameter beig measured. LENGTH MEASUREMENTS Legth measuremets require some scale of compariso for reportig the legth determiatios, ad for this measuremet to be meaigful, the uit of measure should be some widely accepted stadard. For example, with the System Iteratioal (SI) uits the Meter is the base uit for the measuremet of legth. For the Uited States Customary System (USCS) of uits, the ich is a derived uit of measuremet. Oe example of a scale for makig liear measuremets is the steel scale as depicted i Fig. 1 below. Figure 1: Steel ruler with Metric (SI) ad Eglish (USCS) scales Whe makig legth measuremets with a ruler it is importat to uderstad the desig of these istrumets to avoid measuremet error. Most specifically the steel ruler that you will be usig i lab has two scales, o oe side SI i cetimeters ad the other USCS i iches. Please ote the smallest subdivisio of the SI scale is 1 mm, while the USCS scale is resolved ito 1/32" icremets. For this lab we will use the SI scale. It is importat to ot measure a object from the edge of the ruler because ofte the ruler s edge does ot mark the zero poit or the edge could be damaged. File: Measuremet Demostratio (MD1)  Istructios.doc 1
2 For the most accurate results whe utilizig a liear scale for legth measuremets, studets are urged to use the followig procedures: a) Lie up oe edge of the object to be measured with oe of the iteral whole ich tic marks (e.g. 2, 3, 10). b) Next, compare the remaiig object measured with oe of the iteral whole ich tic marks (e.g. 2, 3, 10). c) Report your measuremet to the earest 1/32 of a ich, beig certai to double check ad verify the correct umber of whole ich icremets (see Fig. 2). Figure 2: Correct aligmet of object to be measured with steel ruler A micrometer is a device for measurig diameters of machied objects or the thickess of a material. Figure 3 depicts a typical micrometer with the idetificatio of various parts. As with ay measuremet istrumet, careful use ad hadlig is essetial to make accurate measuremets ad avoid damage that reders it icapable of performig its iteded fuctio. The correct procedures for usig this istrumet are as follows: a) Be certai the lockut (lockig screw) is loose prior to makig ay adjustmets with the thimble. b) Rotate the thimble i a couterclockwise directio (whe viewed from the ratchet ed) util the avil ad spidle are separated a distace greater tha the legth or thickess to be measured. c) Place the micrometer o the object to be measured ad rotate the ratchet kob i the clockwise directio util both the avil ad spidle are i cotact with the object. Cotiue tighteig the thimble util the ratchet screw frictio joit begis to slip. Be careful to tur oly the ratchet screw while makig this adjustmet. Do ot tur the thimble itself as this may result i damage to the istrumet! d) After closig the spidle to the correct positio, set the "lockut" ad remove the micrometer from the object. Read the Verier scale to determie the fial measuremet to the earest i. Procedures for readig the scale are to look at the tic marks o the "sleeve" that are closest to the "thimble." The major marks idicate 0.1 icremets while the fiest tic marks idicate ich icremets. Add to the sleeve measuremet the umber of ich icremets as read from the thimble at the lie o the sleeve parallel to the axis of the sleeve. Figure 4 below provides a example. File: Measuremet Demostratio (MD1)  Istructios.doc 2
3 Figure 3: Idetificatio of micrometer parts Figure 4: Readig the Verier scale o a micrometer DESCRIPTIVE STATISTICS Statistics are parameters used to describe the magitude ad variability that occurs withi a sample of observatios from a populatio. The most commo statistical parameters that egieers ecouter are the mea ad stadard deviatio of a populatio. Whe utilizig these parameters to describe a populatio, we must make a simple assumptio, ad that is that the populatio beig described is distributed "ormally" (Gaussia distributio). You may have ecoutered the term "ormal" i your prior academic career. As ofte times grades may be assiged i accordace with a "bell shaped" curve may studets receive a letter grade of "C" while a sigificatly small proportio of studets get a letter grade of "A." Figure 5 depicts a File: Measuremet Demostratio (MD1)  Istructios.doc 3
4 "stadard ormal distributio." Please ote that two parameters are used to describe the shape ad locatio of this curve, ad they are "mea" ad "stadard deviatio." The "mea" describes the positio of the curve o the horizotal axis, while the "stadard deviatio" is a idicatio of the spread of the distributio. If the horizotal axis is scaled i icremets of 1s, the the cetral portio of the curve from 1.0 to 1.0 cotais 68% of the etire area uder the curve. Correspodigly, as we move to +2s ad +3s, the correspodig area uder the curve is 95% ad 99.7% of the total area, respectively. Figure 5: Stadard Normal Distributio While the Greek characters µ ad σ are used to deote the populatio mea ad stadard deviatio, respectively, it must be recogized that the true values of either parameter may ever be kow. I reality we must sample the populatios, makig measuremets of the parameters of iterest. From this data we ca the estimate the values for mea ad stadard deviatio, ad as these calculated parameters are ot exact, we adopt ew variables to represet these values. The ew variables x ad S are used to deote estimates for the mea ad stadard deviatio, respectively, as calculated from a sample of the populatio. Figure 6 below illustrates the relatioship of the variables i questio. Sample Populatio Mea x µ Variace S 2 σ 2 Stadard Deviatio S σ Figure 6: Descriptive statistics variables File: Measuremet Demostratio (MD1)  Istructios.doc 4
5 To calculate the mea from a sample of the populatio, you simply sum the measuremet values, ad the divide by the umber of observatios as show i the equatio below, X = i= 1 x i where x i is the observatio i's value, is the total umber of observatios from the populatio sample. As you may recall from our previous discussio, the stadard deviatio is a measure of the spread of the ormal distributio. We estimate this value, agai usig the same sample observatio values as before, with the followig equatio, S x xi = = ( xi x) i 2 i= 1 i= 1 i= 1 ( ) Please ote that we divide the sum of square of differeces betwee the idividual observatio values by 1 to get a ubiased estimate of the variace (square of the stadard deviatio). At this poit you might be askig why divide by 1 versus by itself? The justificatio is that we wat a ubiased estimate of the variace, ad by usig 1 we slightly overestimate the variace. The fial step is to take the square root of S 2 to get the stadard deviatio of S. 2 File: Measuremet Demostratio (MD1)  Istructios.doc 5
6 Laboratory Procedures ME 101 Part 1: Measuremets of the washer 1) Determiig the ier diameter, outer diameter, ad thickess of washers a) At each statio there are 10 washers, each of these washers have a idetifier o them. It is imperative that you ote the idetifier with each measuremet that is take. Make sure to record each measuremet ito the proper sectio of the provided table. b) Usig the ruler measure the I.D. (ier diameter) ad O.D. (outer diameter) of each of the washers ad record the values. Make all of iitial measuremets i iches. c) Usig the micrometer, measure the thickess of each of the washers ad record the values. The micrometers used i this lab measure i iches. 2) Determiig the masses of the washers a) Usig the digital scales fid the mass of each washer i grams. b) Determie the mea ad stadard deviatio of the 10 mass measuremets. 3) Homework: Calculatig the desity of the washers ad comparig it to a expected value a) Determie the mea ad stadard deviatio for the I.D., O.D., ad thickess for the set of 10 washers. You may use your calculator or a computer program (such as Microsoft EXCEL) to help calculate the mea ad stadard deviatio. However, do each calculatio loghad at least oe time ad iclude it with your lab report. b) Usig the I.D., O.D., ad thickess measuremets, calculate the volume of each washer i cubic cetimeters. c) Determie the desity of the te washers usig what you kow about each washer s mass ad volume. (Hit: desity = mass/volume) d) Determie the mea ad stadard deviatio of the washer desities. Part 2: Homework: Cocludig Questios 1. Explai i your ow words what stadard deviatio is ad why it is useful whe describig experimetal data. File: Measuremet Demostratio (MD1)  Istructios.doc 6
7 2. What rage for mass of the washers would you expect 68% of the samples to fall ito? What about 95%? (Hit: use the stadard deviatio calculated for mass measuremets) 3. Do you thik that a sample populatio of two washers would have bee adequate to geerate the descriptive statistics for the etire populatio? Explai 4. For what reasos would someoe geerally NOT wat to use the very ed of a ruler for takig measuremets. 5. Assume that the actual desity of the washers is 7.81 g/cm 3. What is the percet error of your calculated mea? What sources of error exist i this experimet? File: Measuremet Demostratio (MD1)  Istructios.doc 7
8 Name Sectio Washer Part I: Washer Measuremets Ier Diameter Outer Diameter Homework: Thickess Volume Mass Desity (uits) ( ) ( ) ( ) ( ) ( ) ( ) Mea Stadard Deviatio Had Calculatios for Stadard Deviatio: File: Measuremet Demostratio (MD1)  Istructios.doc 8
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