UNC Uncerainy revised Augus 30, 017 Learning Objecives: During his lab, you will learn how o 1. esimae he uncerainy in a direcly measured quaniy.. esimae he uncerainy in a quaniy ha is calculaed from quaniies ha are uncerain. 3. use Origin o perform calculaions, make graphs, and perform linear regression. 4. look a a graph wih daa and a model and deermine if he daa suppor he model. (You will urn in wo workshees and wo graphs for his lab. he workshees (UNC Workshee and Error Analysis and Propagaion Exercise) are included in Appendix IX of he lab manual. he maerial for his lab is due one week from he dae of your lab a 6:00 PM.) A. Inroducion he purpose of his laboraory is o accusom you o recording measuremens. You will measure mass, ime, lengh, and force. Besides learning some basic measuremen echniques, you mus realize ha all measuremens possess some uncerainy in heir values. When you make a measuremen (x), you mus esimae he absolue uncerainy (δx) in your measuremen. his quaniy is based upon how well you hink ha you could make he measuremen. he absolue uncerainy may be a funcion of he qualiy of he measuring insrumen, he naure of he quaniy being measured, he abiliy of he individual making he measuremen, and he condiions under which he measuremen is made. oo ofen we assume ha he absolue uncerainy is based solely on he resoluion of he measuring insrumen. he resoluion is he smalles graduaion on a scale or he las decimal place in a digial readou. he absolue uncerainy ofen depends on several of he facors above. here is no prescribed formula of how o calculae he absolue uncerainy for a measured quaniy. For his reason, i is necessary for you o discuss how you deermine he absolue uncerainy of a measuremen in he Procedure paragraph of a laboraory paper. aking he measured value and he absolue uncerainy ogeher, you repor he measuremen inerval (x ± δx). his symbol x ± δx represens he inerval in which we have confidence he measured value lies. Remember o always repor he absolue uncerainy (δx) ogeher wih your bes esimae of he measured value (x). here are rounding rules ha guide us in reporing he number of digis in he answer. For example, we would no repor a number such as.1789345 meers if we were really only sure of he reading o wihin 0.01 meers. Insead, we would repor he value as.18 ± 0.01 meers, i.e., we esimae ha he value lies somewhere beween.17 and.19 meers. In Physics laboraories, we will adop he simple rule ha he absolue uncerainy of a measuremen is repored o only one, or a mos wo, significan figure(s), and he measuremen is hen rounded o he same decimal place as is absolue uncerainy. For example, 0.014 ± 0.0000 kg is properly rounded. he absolue uncerainy is repored o one significan digi, bu ha digi lies five decimal places o he righ of he decimal mark. herefore, he measured value is repored all he way ou o five decimal places. Remember ha he measured value plus or minus he absolue uncerainy, e.g., 1 Uncerainy
0.014 ± 0.0000 kg, represens he measuremen inerval. See Appendix V, Secion D. Relaive uncerainy is a measure of he precision of he measuremen. If a measuremen has a high degree of precision, repeaed measuremens (using he same equipmen and echnique) will produce abou he same resuls. We calculae he relaive uncerainy by dividing he absolue uncerainy of he measuremen by he value of he δ x measured quaniy,. For example, he relaive uncerainy of he value above is x δ x 0.0000 kg = = 0.00 = 0.%. x 0.014 kg Noe ha, as wih absolue uncerainy, we will repor he relaive uncerainy o one or wo significan figures. We will also normally repor he relaive uncerainy in percen form. When you make a measuremen, here is always some error presen. Errors can be divided ino wo caegories: sysemaic and random. Sysemaic errors can cause measured values o be consisenly eiher oo high or oo low. Sysemaic errors affec he accuracy of a measuremen. he accuracy of a measuremen indicaes how close i is o a rue or given value. Random errors are posiive or negaive flucuaions ha cause a measuremen o be oo high someimes and oo low a oher imes. If we ake enough measuremens, he effec of he random error normally averages ou o nearly zero. See Appendix V, Secion A. B. Apparaus Each group member will make measuremens of lengh, diameer, mass, ime, and force of a simple pendulum. You will measure he lengh of he pendulum using a meer sick, he diameer of he ball wih he vernier calipers, he mass of he ball wih an elecronic balance, he period of he pendulum (how long i akes o complee one swing) wih a sopwach, and he weigh (force of graviy on) of he pendulum wih a spring scale. Make he measuremens independenly of oher group members. Do no be influenced by heir resuls. C. Procedure Se up a able in your noebook wih a ile of Lengh and columns for he following: rial number, person who did he measuremen, and measuremen value. ake one measuremen and record he value in he able. Esimae he uncerainy in he measuremen and record i along wih your reasoning for ha esimae in your noebook. Have each parner ake a measuremen and esimae of uncerainy wihou elling each oher wha he measuremen was. Alernae aking measuremens unil he group has aken a oal of en measuremens. Add your parner s measuremens o your able. Repea his procedure for measuremens of diameer, mass, period, and weigh. Noe ha each lab room will only have one balance; you do no need o ake he measuremens in order. D. Analysis Calculae he mean, sandard deviaion, and sandard error for each of he five measuremens. See Appendix V, secion B.1 for how o calculae hese values. Compare he sandard errors wih your esimaes of he uncerainies. Were your esimaes much larger, much smaller, or essenially he same as he sandard errors? Can you explain any differences? Uncerainy
E. Propagaion of Uncerainies and Discrepancies he heoreical relaionship beween he period and lengh of a simple pendulum and he acceleraion due o graviy g is = π. (1) g hus, if we rea our pendulum as a simple pendulum, we can find g by g = 4π () Apply Eq. o your mean values of lengh and period above o find g. Since you are uncerain abou boh and, you are uncerain abou g as well. rea he sandard errors in your values of lengh and period as your uncerainies (δ and δ) o find your uncerainy in g (δg). You should review Appendix V, Secion C. We can find δg by adding in quadraure our uncerainies in g due o (δg ) and (δg): δ = δ + δ (3) g g g We have wo mehods of calculaing δg and δg: he derivaive mehod and he compuaional mehod. E.1. Derivaive Mehod (PHYS115 sudens may skip o secion E.. a his poin.) We can apply Eq. 5a from Appendix V, Secion C.1. o Eq. above o deermine δg and δg: and g = = 4 4π δg = δ (4) δg δ π δ g δg = δ = 4π δ 8π δg = δ 3. (5) Apply Eqs. 3, 4, and 5 o your daa o deermine an esimae of δg. E.. Compuaional Mehod Alernaively, we can direcly calculae how much g changes when and vary by heir uncerainies by applying Eq. 9 from Appendix V, Secion C.1. o Eq. above: and ( δ, ) (, ) ( + δ ) δg = g + g δg = 4π 4π δ g 4π δg = δ (6) (, δ ) (, ) = g + g δ = 4π 4π g ( + δ ) δg = 4π ( + δ ) 1 1 (7) Apply Eqs. 3, 6, and 7 o your daa o deermine an esimae of δg. Repor your value of g as a measuremen inerval (x ± δx). E.3. Discrepancies he difference beween your experimenal value and anoher experimenal or heoreical value (he acceped value) is he 3 Uncerainy
discrepancy. he discrepancy should never be quoed as an error. If a discrepancy is no much larger han a resul s uncerainy (say, no more han wo imes as large), he resul is considered o be consisen wih expecaions. Calculae he discrepancy beween your measured value and he acceped value. For his course, we will use (9.81 ± 0.0) m/s as he acceped value for g. If possible, offer some explanaion for discrepancies ha are larger han he expeced uncerainies. For example, suppose you perform an experimen o measure he speed of sound and find ha your measured value is higher han he acceped value by more han wo imes your esimaed error. You may jusify his discrepancy by menioning ha he unusually high humidiy and emperaure may have affeced your resuls, if his is in fac he case and if hey migh have had he appropriae effec. Ideally, your explanaion should have some physical basis or describe a limiaion in your echnique wih a clear line of reasoning as o how i could have caused he discrepancy, and he explanaion should be quaniaive. F. Origin Exercise Asronau Dr. C. C. aylor was seleced o go on he NASA mission o Europa. One of he experimens Dr. aylor performed while on Europa was o drop a seel ball bearing from several differen heighs and measure he ime i ook o reach he ground. Dr. aylor sen his daa se o Glenn Research Cener where you are assigned he ask of analyzing his daa se o deermine he acceleraion due o graviy on Europa s surface. Your friend Arisole ells you ha you should use he equaion h = aa (8) o analyze he daa where h is he heigh of release, is he ime of fall, and aa is he acceleraion due o graviy. Uncerainy 4 Your friend Newon ells you ha you should use he equaion 1 h= an (9) o analyze he daa where an is he acceleraion due o graviy. Dr. aylor s daa is rial Heigh h (± 0.01 m) ime (± 0.01 s) 1 1.45 1.51 1.30 1.4 3 1.0 1.34 4 1.00 1.3 5 0.90 1.16 6 0.75 1.07 7 0.60 0.97 8 0.45 0.83 9 0.30 0.69 10 0.15 0.48 F.1. Direc Calculaion One way o analyze his daa is as a se of 10 measuremens of he acceleraion. Open up Origin. Go o he columns menu, choose Add New Columns, and add 4 new columns. Righ-click on he op of firs column and choose Properies. ype in Heigh (m) in he long name area. Change he plo designaion o Y and click he Nex buon. Choose yes o he quesion Do you wish o auomaically display he Column Label? Change he nex long name o Uncer. in h, and change he plo designaion o Y Error. Click on he Nex buon. Make he nex plo designaion X, and long name ime (s). Click on he Nex buon again and make he nex plo designaion X Error and long name Uncer. in. Make he nex long name Arisole s acceleraion (m/s) and he las long name Newon s acceleraion (m/s^). Click OK. Ener Dr. aylor s values for h and ino he appropriae columns. o pu he uncerainies ino Origin, selec he column dh, go o COLUMN\SE COLUMN VALUES. In he box ha pops
up, change Row(i):From Auo o Auo o for rows 1 o 10 and change he defaul equaion in he box below col(b)= o 0.01 and click OK. Use he same procedure o se column D o 0.01. Now we re ready o sar calculaing aa and an. From Eq. 8, we can see ha aa = h/ (10) and from Eq. 9, an = h/. (11) Raher han doing 0 calculaions on a calculaor, we can have Origin do he calculaions for us. Selec he column E and bring up he se column values box. Se he range of rows o 1 o 10. A his poin, you wan o make Origin calculae he whole new column for Arisole's version of acceleraion for you (aa = h/). o have his done on whole column by origin, ype "col(a)/col(c)" o he equaion box hen click "OK". Selec he column F and se is firs 10 rows values o *col(a)/col(c)^. Now we are ready o calculae he mean, sandard deviaion, and sandard error in our experimen. Righ-click on column E and selec Quaniies, check Mean sandard deviaion and SE of Mean, and hi ok. Record he mean (Mean(Y)), sandard deviaion (sd(yer±)), and sandard error (se(yer±)). Repor a A as a measuremen inerval. Go hrough he same procedure on column E o find he mean, sandard deviaion, and sandard error of a N and repor a N as a measuremen inerval. (Do no prin ou he ables produced by SAISICS ON COLUMNS; jus record he appropriae numbers in your laboraory noebook.) Wha conclusions can you make as o which model (Arisole s or Newon s) beer describe he daa? Wha is he reasoning behind hese conclusion or reasons why you can make a conclusion? F.. Fiing a Model o Daa Origin is a powerful graphing and fiing program. We will now use ha power o more precisely es he validiy of our models. F..a. Arisole s Model Eq. 8 implies ha if we plo h on he y- axis and on he x-axis, he slope of he bes fi line will be aa. Furhermore, if Arisole s model is valid, he daa should randomly scaer abou ha bes fi line. In Origin, make sure no columns are seleced. Go o PLO / SCAER. Selec columns A, B, C, and D as your Y, yer, X, and xer columns respecively. (If he xer check-boxes don appear in he Plo Seup window, righ-click on he window and selec X Error Bars. ) Click ADD and hen OK. We now have a fairly ugly looking graph; we need o prey i up o make i informaive o ohers. Double-click on he Y Axis ile and change i o h (± 0.01 m). (o inser he ± symbol, hi Crl-m o bring up he symbol map.) Change he X Axis ile o (± 0.01 s). Double click on a daa poin. A Plo Deails window will appear. Make sure he firs se under Layer 1 is seleced (Daa 1: (X), h(y).) Change he size of he daa poins o 3 poins and click OK. (Wih 8-poin daa markers, he daa markers are bigger han he error bars!) Use he ex ool o give he graph he ile Acceleraion Due o Graviy on Europa- Arisole s Model. Use he ex ool o add your name and your parner s name o he graph. Make sure he legend doesn overlap he ile. Now we re ready o fi a line o our daa. Go o ANALYSIS / LINEAR FI. A red line will appear on your graph (he bes fi line) and some numbers will appear in he resuls window (usually a he boom righhand corner of he Origin window). Copy he linear regression daa and pase i ino he graph. You will probably need o make he 5 Uncerainy
he uncerainy in ( δ ) is no he square of he uncerainy in ( δ ). We can apply Eq. 5a from Appendix V o find δ : d δ = δ d δ = δ (1) So each of our values of will have a differen uncerainy associaed wih i. (Alernaively, we could use he compuaional mehod o find Figure 1: Sample graph of Arisole s Model. fon size smaller. Noe ha Origin does no handle significan figures properly; you will have o edi hese values manually so ha he number of significan figures is appropriae for he errors ha Origin calculaes for each value. Your graph should look somehing like Figure 1. Save your Origin projec o a folder you produce on he lab server and prin ou a copy of he graph for each member of he parnership o include wih he workshees. Repor a A from your bes fi as a measuremen inerval. F..b Newon s Model Eq. 9 implies ha if we plo h on he y- axis and on he x-axis, he slope of he bes fi line will be ½aN; i.e., an will be wice he slope of he bes fi line. Furhermore, if Newon s model is valid, he daa should randomly scaer abou ha bes fi line. In order o creae his plo, we need o creae wo new columns: one for and one for δ. Creae wo more columns in your Origin projec and make he firs of hese new columns he column and se as a plo designaion of X. Use he Se Column Values command o se he values of hese cells o. ( δ ) δ = + δ = + δ + δ δ = δ + δ δ δ. (13) As you can see, boh mehods yield he same resuls when δ <<.) Use he second new column as your δ column. Use he Se Column Values command o se he values of hese cells o δ. hen creae a new plo wih h as your Y variable and as your X variable. Don forge o include error bars! Make your plo prey and find he bes fi line and he bes fi line parameers as above. Save your Origin projec o a folder you produce on he lab server and prin ou a copy of he graph for each member of he parnership o include wih he workshees. Calculae a N and δ an from he slope of he bes fi line and repor a N from your bes fi line as a measuremen inerval. F..c. Conclusions Sudy your wo graphs. Which model more closely fis he daa, Arisole s or Newon s? Wha is your evidence ha one is a beer fi han he oher (or hey are equally good)? Wha value of he acceleraion due o graviy a Europa s Uncerainy 6
surface would you repor o your supervisor a Glenn Research Cener based on Dr. aylor s daa? Jusify his choice. Your coninued employmen depends upon i! 7 Uncerainy
his page inenionally lef wihou useful informaion. Uncerainy 8