ࡱ> C8 9  !"#$%&'()*+,-./01234567A:;<=>?@BLDEFGHIJKMQNOPRRoot Entry F8fmc]pc8Workbook Y_VBA_PROJECT_CUR"~ipc]pcMBD0009951CF~ipc~ipcMBD00190C0EF~ipcqpcOle Equation Native Ole F  !"#%'()*+-./0123456789:;=>?@ABCDEFGHIJKMNOPQRSTUVWXYZ[\]^_`abcdefghijklmopqrtvwxyz{|}~T@ .4MEOW dV_dD="V"D=14(2D)LdV_dL="V"L=14D 26F!Calculations!Picture 15L'; <PIC  LMETA  CompObj $fEquation Native &U';  ?  .1  ``&  & MathType-) k -Times New Romank~wWw -2 @VolumeY 2 Vo 2  Do 2 / Lo 2 Vo 2 hDo 2 o Vo 2 h Lo Times New Romank~wWw  - 2 vp 2 D 2 ~ LSymbol b~wk~wWw - 2 = 2 = 2 p= 2 y 2 Ky 2 y 2 y 2 Ky 2 y 2 p/ + 2  y 2 K y 2  y 2 'y 2 K'y 2 'y 2 J y 2  y 2 { y 2  y 2 Jsy 2 sy 2 {sy 2 syTimes New Romank~wWw  - 2 21 2 4 Times New Romank~wWw - 2 r 2p 2 ^2p 2 2p 2 B1p 2 2pSymbol  b~wk~wWw  - 2  p 2 p7s 2 A 2 hB 2 p s 2   2 h  2 p s & "System  - FMicrosoft Equation 3.0 DS Equation Equation.39q9< Volume=V=14D 2 LU v ="V"DU D () 2 +"V"LU L () 2 [] )12'#xME @\pMechanical Engineering Dept. Ba= ThisWorkbook=!+#8X@"1Arial1Arial1Arial1Arial1Arial10Arial1.Times New Roman1.0Times New Roman1.Times New Roman1.Times New Roman1.Times New Roman1.Times New Roman1.Times New Roman1.Times New Roman1Symbol1.Times New Roman1.Times New Roman"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)0.0 0.00000 0.0000 0.000 0.0000000 0.000000 0.00000000 0.0E+00 0.E+00 0.0%                + ) , *         "   (@@  (@  ( @ "   (@  (  ( "8  # # !  h     h@@+  h @+  (  h@+  h +   h@ +  h +  #8@@ "8@  (@   h@@+  h@+  #8@ "<  (  (  h+ "8 !x@+ "x+  h+  h@ + "x +  h + !x@ +  h +    ! "  (   ( " "\,  ! #x@@, #|@,  "x@,  h @, #x@,  h@ ,  #| ,  "x , #| ,  h ,    X+ X+  "< !< #8@  "  !  I  x@@+  x@+  x @+  x@+  x+  x +  x@ +  x +  x + "   x@@    x@    x @   "8 #    "  "\,  "X, `0 CalculationsdV_dD:+dV_dL:-dVdDnum:- dVdLnum:. mean_D:mean_L:! measd_L;n_D:n_L:smu_D: smu_L: stdev_D:stdev_L:t95pct: tau95:!Uncert_D:(Uncert_L:)Uncert_V:/Vvar_D:Vol:0`i )O$"49jaccA5i@=a49jaccA5\#Q0|/xY}lEw+;@)i/.(6X)QKyՖG&Clb D(R(Xߛy{rJfyy3z,,.hi$t p\^Iyg^m@, SM7hڼ=IbDK#T~߮B#/O]~PWI~gi,ރ"DwK"%ڀ)ߕޔxgoHw(t/֗^Y$cFOiIH(;~~Odz */;B{?HO'|aSiC? u \+~W?I\쯥?I\CYRGMsc&ĜL c^+j+{/n5]#MHJ˵HcΝ(ǃ̥)O߂c,KWi1k-__WZѸ.;?6#a#u5㧃[ Gb}!U ַ%йoCnw4SvCs&wIdg#EnNH Y/ k{͐J%k8I_WNe[_8 I氬k&#yk¢i"aOsR^^k~kk`е?,l&}VMX` Pn4K@BRЗtUDĸyl߄~Ѭ k{,PKf gsϟ295ObEr_4q׀ܓ-vp|s;87 x_hmzSV|l TYs33a3E~׊ә|gCA,g㸳sMMso(&0s\ A72E8,8|ujw/hbA7q`y9pcmz~M/j_hLyC ?2ʳvSaL;S [i{_}vוcX0X.qpƫ0nC!O LN#̀ɚoV/voz"7@Ὲ8Ѧ%9rKjȿgoٖÿ0݄c(<%fY)r/9Wsq.saq6*8 A*q.5ҰF0wY9mrbi:-Ҡc{'8q,།9{m8{_Ӆ ̹Z u F?zݍ!m٣t3*^Wp_ sx;8#|&i<=N̝fM .# ΛSv??Ul NnlYy3sv*8 w1`ӳ<[+X^,=E1YN,=E1YN,E [*ʋޢ8ˉޢ8ˉ9Xj𸋥iYw{0'g3s.Wp+Ύs6islq\8[roD!wD/x|&{\Z_yzzCZnFqݍCd`.lq-WalϺcL__3ά$" >t))w@=o >t))8D=xXklTE>{[X*VZҖ6*k EͶ*.-)饉1 4$Cc!4$4!6 ?P1$E3{^K-Μy|39^Z0aڋ L_g{/(P5` ;zP4Dp|yeh 86K$v6 ʚ(MV[䋶2Uf$fdݵYsgl@ʃX;GR@̍sĘLo޶i]wG+nPbm"cʐj3m#^v۲]b\#ye[k{|} [Ku㭝5Z=޼"}co]uw~}u4 ޵SYy3}K\><2!\^fA=I\ %~K|GaTAfvڣKjP3&9s =GU:Z-a)Wi}`Ew(*˕J;O!)K| U;kX:p\>Z#*>"LT} :dG1W*B8w}:}} IJ82=Fo|%_8T%;ƀC^,]j1uZHt j5P{K@+ c*~7#WeX Xp|1(U-twnk6x3~f#݁O/31e?$狱NzBԉfgz+!ɮîEcǍ)_?v@ !`R!wy,G#>c?Cٮd˲2~-LSfC+tBVаl.ͰRB(tLo`x&d5yQwuτMhx0?}a|}pԤVF'#Vk4s΄w;g"%>B8٤Fv9^i;i|$Ҙ&J\'r'g0\偮v Hkټq]Hl0xQkF pLqvW?|X_K,ֈuoJNa,mp?l,&̓yY(3a>y82azQ+=f sp|,9at10 . ̂ {&f =>7 0M7p;60+pa&,uĬTwҽBwc(w$^WD#|\?`O}e:H^{*^pq %9uz^xq2oniUo~nExENWG+#yš&ZUY k迪"n{\E}zXVB*W-q*|*" R``nքkK=(ÔXws[1fvcy&O>ӹR9kg1cVA*gqVSO/}T]z?= 7o7tĿX/`j)>2>c>kW݊$ʱ紘i5+/c߲w5%sZ33gކs;ڰЁY<H se ia=.8a{k#)5֐kcKNVçuF=؊@jl؋e` =Nq`=e؊>KR N16@|߬v6|aji3+m'cS;@zdSOg$tX c*Oda3  @|@3 x^$The measured values are given below.(D)meas(L)meas(mm)smu_Dmmsmu_L cm equals Solution:Vol = MeanVol +/- (s)volmean_Dmean_Lstdev_Dstdev_LdV_dDdV_dLVVol+/-%PROPAGATION OF ERROR ANALYSIS EXAMPLEdVdD =dVdL=(mm^3)Vol =(m^3)1The smallest measuring units associated with the !measured diameter and length are:INSERT:NAME:DEFINE3These values are calculated using referenced cells.cells./place an apostrophe before the formula, and hit4return. Voila - you can easily see the formula now.5Be sure to check that the referenced cells are listed correctly."V / "D"V / "LThese values have beennamed "measd_L" using(mm) =AVERAGE(C22:C26)(mm) =STDEV(C22:C26)1Numerical Differentiation Using a Function ModuleAnalytical Differentiation0NOTE: To show the formula of a particular cell,6These values are calculated using the referenced named1 var_Dn_D =COUNT(C22:C26) - 1(mm2) =C30_D^2t95% Table valuen_L =COUNT(measd_L) - 1!File: Prop_of_Errors_Cylinder.xlst95stdev_D/sqrt(n_D) = t95smu_D/sqrt(12) = Uncert_DUncert_LUD UL UV Uncert_V1(mm)=MAX(t95pct*stdev_D/SQRT(n_D),smu_D/SQRT(12))Scenario: You are asked to determine the volume, V, of a cylinder using measurements of the cylinder's diameter, D, and length, L. The answer is to be reported as a mean value and the associated uncertainty. U.Vanal =Vnum =Non Dimensional Analysis!Uncertainty Magnification FactorsUMFD =UMFL =#Uncertainty Percentage ContributionUPCD =UPCL =$ =dV_dD^2*Uncert_D^2/Uncert_V^2*100 =dV_dL^2*Uncert_L^2/Uncert_V^2 =dV_dD*mean_D/Vol =dV_dL*mean_L/Vol@Notice the final answers are given to only 3 significant digits. Uncert_V =(cm)(cm) =AVERAGE(measd_L)(cm) =STDEV(measd_L)(cm2) =stdev_L^2t95stdev_L/sqrt(n_L) t95smu_L/sqrt(12) 8(cm)=MAX(t95pct*stdev_L/SQRT(n_L),t95pct*smu_L/SQRT(12))A=ROUND(V(mean_D,(mean_L*10)+Uncert_L*10)- V(mean_D,mean_L*10),-3)==ROUND(V(mean_D+Uncert_D,mean_L*10) - V(mean_D,mean_L*10),-3)$=ROUND(SQRT(dVdDnum^2+dVdLnum^2),-3)=ROUND(V(mean_D,mean_L*10),-4)3(mm^3) = ROUND((1/4)*PI()*mean_L*10*(mean_D)^2,-4)B(mm^3) = ROUND(SQRT((Uncert_D*dV_dD)^2+(Uncert_L*10*dV_DL)^2),-3))(mm^2) = ROUND((1/4)*PI()*(mean_D)^2,-2)1(mm^2) = ROUND((1/4)*PI()*2*mean_D*mean_L*10,-2)Notice the diameter measurement contributes ~60% of the uncertainty. This might be expected, since the UMFD is greater than 1, indicating the uncertainty of the diameter will be magnified as it propagates through the data equation.lmb$ %p_%%'&D'(])*"x+_,-' @ C5@N'P  dMbP?_*+%'?(?)?M\\Dunnlab\HP LaserJet 8000 Seri?dX,Letter.HP LaserJet 8000 Series PCL 62)xe.DauC7\"-1",0ĺT]:1%aAx,3Zs[TǀVpg8%~@ CC ]h!E{]}B@Rf<-l5چ܂R|ɕI?49G@j݇$x 2m>|Ю;Σ6QAᾱƭiĆ᳧)>3)>;B32<$EI k@JF9`3zz 5w7Q:E'ؼbnH^ >@b=k" P c뮲=*?6;8uC#?Nh LwNbA-^ j"AX,??U} } $ } $ } $ } I }  } $ }  }  } $ } I } $ C @        f"ffffffffffffff 4 g>hhhhhhhi        ! !N "~ #? $ @`@:@ "~ #@ $ #4@) D    @_@9@ @`@9@ @_@9@_@9@%% & & %%   '$( )) ! !N *%+ ))@`@D :@D  ,- ))@_@D 9@D  ))@`@D 9@D  ))@_@D 9@D  ))_@D9@AD )) )) )) . #/_@2 %B 0& . /3333339@ #B 0O  1*  22( )) 3 #4op@ %B 5'6 3 4m%? #B 5P)  *  77+ )) 3,V03333@ D W/6 3,V? 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AU /T; /"<R/G AB<CCC C@@AA /AZ /&& /"M8/ n@ A- "CCAA /  / HW 0U 0"C0G}HA+-AC CA 0AY 0&& 0"90 o}HA. ###C BA 0  0 AX 2" 2I?2J}HA3D0 2K2J A3D/ 2L 2I@2 M}HA3 D0 2 K2 J@ A3 D/ 2 L3N%3O&Kj?3 D2& .> 3P)3QI'-?2 @D2& .> 3R3N%3 O&Kj?2D2 & .> 3 P%3 Qz.?2D2 & .> 3 R5S 5]L 5^_ 6cde 7%%% 8jA8jjjj :B <#C'<[˜@%CCC <[J =#D+=[ȿ$?0 CC C =[K =]] =^_ >`ab ?E ?`ab < RF.(v,~D& Gg@ A @B  @`ab A#F3A\?vb%?<CCC A\H A`ab B#G8B\346?="CC C B\I Bcde (cT (<(    RA ?+@Picture 15  ]J  phv Equation.2 ~v $ <A ?+@?P*@/ ]J$` hv Equation.3 td RUZ* (3   pw] (`  Liv@A@Text 14RmR]ivt  < Diameter, D<E   LLiv  @Text 17UUn]Liv  < Diameter, D<B   Liv  @Text 18Z]iv  < Length, L<3 phB  0D@]`phB  0D@]`P!|tB  <D@]`0"vnB B 6D@n]`"zb   B v @@~t*] `$vnB  6D@h]`l$><d *=B  88567 Sheet1 VBAupc0pcThisWorkbook,Sheet1<Module1LJ(SLSS<N0{00020819-0000-0000-C000-000000000046}@(%H@0r6xAttribute VB_Name = "ThisWorkbook" Bas0{00020P819-0C$0046} |GlobalSpacFalse dCreatablPredeclaIdTru BExposeTemplateDerivBustomizD2'#xME (SLSS<N0{00020820-0000-0000-C000-000000000046}@(%H@0r6xAttribute VB_Name = "She@et1" Bast0{00020820- C$0046} |Global!SpacFalse dCreatablPre declaIdTru BExposeTemplateDeriv$Bustom izD22RxME (6 L< `k@ I `I  %`@XXp@l X$@h @"X @8H @8 @ ( @h @P @ @ @ `r6$*\Rffff*063e9ece82*\R1*#229*\R1*#c4 B  8 (@ &h (  **5( 81This function determines the volume of a cylinder!D = diameter of the cylinder (mm)L = length of the cylinder (mm) t!V = volume of the cylinder (mm^3) $%"'"9/$Calculate the volume of the cylinderiew of "  'iAttribute VB_Name = "Module1" Function V(D, L)  'This fldetermines th@e voluof a cyli@nder rDdid D`$ (mmaL'@length%Vg%n K^3MwPi)ApplicBa.Pi('Calcula!(#8(1 /@ 4) * ** (D ^ 2L End yrU~| i\h$ hW__SRP_2n __SRP_3 sy_VBA_PROJECTu dirHXh5k8XH((l\,lX hHkrU ,`aq//nam  *\G{000204EF-0000-0000-C000-000000000046}#4.0#9#C:\PROGRA~1\COMMON~1\MICROS~1\VBA\VBA6\VBE6.DLL#Visual Basic For Applications *\G{00020813-0000-0000-C000-000000000046}#1.3#0#D:\Program Files\Microsoft Office\Office\EXCEL9.OLB#Microsoft Excel 9.0 Object Library*\G{00020430-0000-0000-C000-000000000046}#2.0#0#C:\WINNT\System32\STDOLE2.TLB#OLE Automation*\G{0D452EE1-E08F-101A-852E-02608C4D0BB4}#2.0#0#C:\WINNT\System32\FM20.DLL#Microsoft Forms 2.0 Object Library*\G{7E961C2D-F62B-46F6-A9B0-08BDDBD0157A}#2.0#0#C:\WINNT\Profiles\samorton\LOCALS~1\Temp\VBE\MSForms.exd#Microsoft Forms 2.0 Object Library.E .`M *\G{2DF8D04C-5BFA-101B-BDE5-00AA0044DE52}#2.1#0#F:\MSOFF97\Office\mso97.dll#Microsoft Office 8.0 Object Library  r6ThisWorkbook033e9ece80ThisWorkbook- Sheet1043e9ece80 Sheet1-Module1063e9ece82Module10XP0dI^Fn34ugn*@)zAbT=NLKUB0& Excel+ VBAWin16~Win32MacVBA6# VBAProjectstdole`MSFormsCOfficeu ThisWorkbook| _Evaluate Sheet1 Module1b(VmD[LcPi^ Application* _B_var_PiVWorkbookk Worksheetl 6  0* pHd VBAProject4@j = r r6 J< rstdole>stdole f%\*\G{00020430-C 0046}#2.0#0#C:\WINNT\System32\STDOLE2.TLB# Automation^MSForms>SFErms3 DD452EE1-E08F-101A-8-02608C4 D0BB4DFM20.DLL#Microsoft 9 \ Ob Library/;C1s00}#0#j0 A7E961C2D-F62B-46F6-A9BZ8BDDBD0157A6@files\samorton\LOCALS~1\Temp\VBE\U.exd=@#.E .`M OfficOfic5y o32DF8D04C-5BFAAiB-BDE5KAA423F:(OFF97\\mso97.dll+p 8.0i@³ThisWorkbookG Th,sWkbokT 2 HB1-,B,!*"B+BSheet1GS@#e't1 qn2 jodulejo|u|A2n FX !bK*mrU~~~~~~~~~i ʟ\NF[     a !) __SRP_0__SRP_1\PROJECTwmVPROJECT  VBAProject ThisWorkbookSheet1Module1F /C:\PROGRA~1\COMMON~1\MICROS~1\VBA\VBA6\VBE6.DLLVBA A i0F 3D:\Program Files\Microsoft Office\Office\EXCEL9.OLBExcel 9a@0FC:\WINNT\System32\STDOLE2.TLBstdole AiP.E .`M C:\WINNT\System32\FM20.DLLMSForms aa`-~+Fz8C:\WINNT\Profiles\samorton\LOCALS~1\Temp\VBE\MSForms.exd 9aapL-[DRF:\MSOFF97\Office\mso97.dllOffice 9aV FFAiPiArU DLpThisWorkbookThisWorkbookSheet1Sheet1Module1Module1ID="{6E4D766C-E2DD-11D2-95DF-00E029225B38}" Document=ThisWorkbook/&H00000000 Document=Sheet1/&H00000000 Module=Module1 Name="VBAProject" HelpContextID="0" VersionCompatible32="393222000" CMG="1D1FC7A849FD4DFD4DFD4DFD4D" DPB="6E6CB4175439A43AA43AA4" GC="BFBD65C6E74639473947C6" [Host Extender Info] &H00000001={3832D640-CF90-11CF-8E43-00A0C911005A};VBE;&H00000000 [Workspace] ThisWorkbook=44, 44, 617, 484, C Sheet1=66, 66, 639, 506, C Module1=88, 88, 716, 547, Z Oh+'0HPd  Jane LebarMechanical Engineering Dept.sofSummaryInformation(DocumentSummaryInformation8 CompObjfMicrosoft Excel@xnc@'@pc՜.+,0 PXx  University of Wyomingul  Calculations  Worksheets FMicrosoft Excel WorksheetBiff8Excel.Sheet.89q