250 I-1 INTRODUCTION PERCENT ERROR AND PERCENT DIFFERENCE

Transcription

1 50 I- INTRODUCTION PERCENT ERROR AND PERCENT DIFFERENCE A percet error hould e calculated whe a eperetal value E copared to a tadard or accepted value of the ae quatt. We epre the dfferece etwee E ad a a percet of the tadard value : E - PE 00 Th forula eld a potve reult f the eperetal value hgh coparo to the tadard value, ad a egatve reult f t low. A percet dfferece hould e calculated whe two eperetal value, E ad E, are copared to each other, ad there o tadard value for coparo. I th cae we epre the aolute value of the dfferece etwee the eperetal value a a percet of the average of the two value: () PD E - E Percet dfferece alwa potve. 00 E - E ( E E ) ( E E ) 00 () THE METHOD OF DIFFERENCE The Method of Dfferece ued whe oe varale eleved to chage equal aout ucceve eaureet. Th ethod eld the average chage the varale per terval. A phcal eaple of uch a cae the tretchg of a prg a force whch creae equal aout ucceve terval. Aother eaple the peed of a dee fallg od eaured at equal te terval. A a cocrete eaple to llutrate the Method of Dfferece, uppoe that we wat to eaure the wdth of oe of a uer of detcal floor tle a roo. Oe ethod to place a eaurg rod dow, eaure a tle, ove the rod, eaure aother tle, ad o o. We could the fd the average of the dvdual eaureet. However, ovg the rod creae the eperetal error. A ore prece ethod to place the rod dow ol oce, ad the to take a et of readg of the poto of ucceve crack (aued to e of eglgle thcke). For eaple, ucceve readg would pa fve tle. Let the readg e a,, c, d, e ad f. There are everal wa whch thee uer could e coed order to eld a gle tle wdth.

2 50 I- A poor ethod would e to ue the forula f - a w 5 The reult would e approatel correct, ut le prece tha t could e, ce we have ued ol two of the readg. A ethod that look etter at frt to calculate the fve wdth, -a, c-, d-c, e-d, ad f-e, ad the average the. The equato for th procedure w ( - a) ( c - ) ( d - c) ( e - d) ( f - e) 5 Cloe pecto of th equato, however, how that t reduce to equato (), o we have gaed othg all of our etra work. The Method of Dfferece ue each of the readg oce, ad o readg cacel out. I order to ue t, we ut have a eve uer of readg. For a odd uer of readg, ether the frt or lat readg ut e dcarded. The, we dvde the readg to two et. I our eaple, et oe would cot of readg a, ad c; et two would cot of readg d, e ad f. We ca get oe etate of the average ug readg a ad d, d - a. Here, the dtace (d-a) pa three tle, o we have dvded three. We ca get two other etate ug the par ad e ad c ad f, e - ad f - c The ethod of dfferece ue the average of thee three etate w d - a e - f - c. ( d - a) ( e - ) ( f - c) I the geeral cae, uppoe that we have ucceve readg, A to A ad B to B, of oe varale. The the average chage per terval gve ( B - A ) ( B - A ) L ( B - A ) () (4) (5) (6)

3 50 I- UNCERTAINTY IN DATA AND CALCULATION I. Ltato o Preco of Meaureet No eaureet eact. The preco of a eaureet lted the ature of the eaurg truet telf, the codto of eaureet, ad the kll of the pero ug the truet. For eaple, the preco of our eaureet of the legth of a oard, ug a eter tck, lted the fact that the allet dvo o the tck a lleter. We ca etate the legth of the oard to a preco of the earet ffth or teth of a lleter etall dvdg the lleter to aller dvo, ut oo we reach a lt. Eve f we were to ue a agfg gla a effort to dvde the lleter to eve aller part, we would evetuall e lted rregularte the ar o the eter tck. A coo ource of error ug a eter tck paralla error. Th error caued the le of ght ot eg precel perpedcular to the tck. For eaple, eaurg the legth of a heet of paper placg the tck dow flat o the paper eal lead to paralla error; t would e etter to place the eter tck o edge o that the ar touch the paper. Paralla error ca occur a tuato; t occur, for eaple, readg a electrcal eter whe the poter ove a dfferet plae fro the cale. II. Aolute ad Relatve Ucertat uppoe that we eaure the legth of a pece of paper a 0.00 c, ug a eter tck. The, after coderg the proce of eaureet, we decde that we ght e error a uch a 0.0 c (oe-ffth of the allet dvo of the tck). We uuall epre th ag that the legth 0.00 ± 0.0 c. I th cae, 0.0 c would e the aolute ucertat the eaureet or the pole aolute error the eaureet. We defe the relatve ucertat a eaureet or the pole relatve error a eaureet to e the rato of the aolute ucertat to the actual eaureet; for our eaple, 0.0 c/0.00 c %. Note that the aolute ucertat ha the ae ut a the eaureet; wherea the relatve ucertat utle ad ofte epreed a a percet. Ucertat ad pole error are o. Ucertat ore ofte aocated wth eaureet. Pole error ore ofte aocated wth reult calculated fro eaureet.

4 50 I-4 III. Ue of Dfferetal to Repreet Ucertate Let the legth of the paper ecto II e called L. The we repreet the aolute ucertat L the ol dl. We repreet the relatve ucertat L the rato dl/l. ce relatve ucertate are uuall rather all, the cocept of a dfferetal ueful deterg how two or ore ucertate coe together whe eaureet are ued atheatcal operato. Th wll e llutrated the et ecto. IV. How Ucertate Data Affect Calculated Reult Coder a forula F(A) evaluated for oe eaured quatt A whch ha a df df ucertat da. If da all, F(A ± da) F(A) ± ( da ) da, where da the dervatve of the forula wth repect to A evaluated at the eaured value of A. Note: Th forula pl the frt two ter of the Talor Epao of F at A. The ucertat df the calculated value for F the df da da. Th ae relato ca e ued for a forula F, f A a calculated quatt. Eaple: Let F(A) A ad A e eaured a.54 ² ± 0.07 ² df da A /(.88 ) df df da da ² F.88 ± 0.0 Now, coder a forula F(A, B, C) evaluated for the depedet quatte A, B ad C whch have ucertate da, db ad dc, repectvel. If da, db ad dc are all, F F F F(A ± da, B ± db, C ± dc) F(A, B, C) ± ( A ) da ± ( B ) db ± ( C ) dc, where the dervatve are evaluated at the eaured value of A, B ad C. The ucertat the F F F calculated value for F the df da B db C dc. Let u ow coder the ot frequet operato we wll perfor calculato: addto, utracto, ultplcato, dvo, ad rag to a power. A. Addto Let e the u of two depedet quatte, A B. ad B. Therefore, d da db. Eaple: (5.0 c ± 0. c) (0.0 c ± 0.5 c) 5.0 c ± 0.7 c For calculato volvg depedet quatte, refer to ecto IV-F.

5 50 I-5 B. utracto Let D e the dfferece of two depedet quatte, D A - B. D D ad B -. Therefore, dd da db. Eaple: (5.0 c ± 0. c) - (0.0 c ± 0.5 c) 5.0 c ± 0.7 c C. Multplcato Let P e the product of two depedet quatte, P AB. P P B ad B A. Therefore, dp B da A db. Dvdg oth de P A B, we fd dp/ P da/ A db/ B. Eaple: (5.00 ± %)(8.00 ± %) 40.0 ² ± 4% 40.0 ² ±.6 ² (.6 ² otaed calculatg 4% of 40.0 ².) D. Dvo Let Q e the quotet of two depedet quatte, Q A/B. Q Q /B ad B -A/B². Therefore, dq /B da A/B² db. Multplg oth de / Q B / A, we fd dq/ Q da/ A db/ B. Eaple: (.00 g ± %)/(.50 c³ ± %).00 g/c³ ±4%.00 g/c³ ± 0.08 g/c³ Note: The geeral rule for addto ad utracto that the aolute ucertat a u or dfferece equal to the u of the aolute ucertate the quatte added. The geeral rule for ultplcato ad dvo that the relatve ucertat a product or quotet equal to the u of the relatve ucertate the factor. pl put: If F A ± B, df da db. If G A B or G A/B, the dg/ G da/ A db/ B. E. Epreo Cotag Ol Power, Multplcato ad Dvo ce power are pl repeated ultplcato, we have, effect ol ultplcato ad dvo. Fro the reult of part C ad D, we ee that we ca pl u the relatve error. For eaple, f A B p C q /D r, the da/ A p db/ B q dc/ C r dd/ D The det of a coe ρ V, where the a, r the radu, ad h π r h the heght. The relatve ucertat the det dρ/ρ d/ dr/r dh/h, where d, dr ad dh are the aolute ucertate the a, radu ad heght repectvel.

6 50 I-6 F. Epreo cotag depedet quatte. Oe ut e careful to ol appl the rule aove to epreo cotag depedet quatte or ele error ca e doule or trple couted. Two quatte calculated fro the ae eaureet are depedet, a are a eaured quatt ad a value calculated fro t. Oe ple geoetrc eaple of the latter cae the calculato of the dfferece etwee the crcuferece ad daeter of a crcle. Coder a crcle wth a daeter eaured to e 0.0 c ± 0. c (0.0 c ± %). It crcuferece C πd. ce π a cotat, t relatve ucertat zero ad the crcuferece wll have the ae relatve ucertat a the daeter, %. The reult C.4 c ± 0.6 c. Ug the rule aove, we would fd C d.4 c ± 0.8 c. The ucertat the dfferece actuall le tha 0.8 c. To calculate the actual value we ut put together the two forula aove, C d πd d (π-)d. ce (π-) a cotat, t relatve ucertat zero ad the dfferece wll have the ae relatve ucertat a the daeter, %. Thu, C d.4 c ± 0.4 c. A eaple of the frt cae would e the rato of the volue of a phere to t urface area. Coder a phere wth a daeter eaured to e 0.0 c ± 0. c (0.0 c ± %). The urface area of the phere A πd. ce π a cotat, t relatve ucertat zero ad the urface area wll have twce the relatve ucertat of the daeter, 4 %. The reult A 4 c ± 4 %. The volue of the phere V /6 πd. ce π/6 a cotat, t relatve ucertat zero ad the volue wll have three te the relatve ucertat of the daeter, 6 %. The reult V 54 c ± 6 %. Ug the rule aove, we would fd V/A.66 c ± 0 %. The relatve ucertat the rato actuall le tha 0 %. To calculate the actual value we ut put together the forula aove, V/A (/6 πd )/( πd ) d/6. ce 6 a cotat, t relatve ucertat zero ad the rato wll have the ae relatve ucertat a the daeter, %. Thu, V/A.66 c ± %. A how the two eaple, whe a epreo cota depedet quatte, t ecear to algeracall plf the epreo efore calculatg the relatve ucertat. G. Cople Epreo Cople epreo a e hadled repeated ue of the prevou reult or the ethod outled at the egg of th ecto (ecto IV). BC Eaple: Let A DE F where B 50.0 ± 0.5, C 0.0 ± 0., D 00 ±, E.00 ± 0.06, ad F 50.0 ±.0 ². oluto I: B C DE F ( 0.0 ) ( 00)(.00) ( 50.0 )

7 50 I-7 BC C DE F D E F - EBC ( DE F) - DBC ( DE F) BC ( DE F) ( )( 0.0 ) ( 00)(.00) ( 50.0 ) (.00 )( 50.0 )( 0.0 ) ( 00 )(.00 ) ( 50.0 ) ( 00 )( 50.0 )( 0.0 ) ( 00 )(.00 ) ( 50.0 ) ( 50.0 )( 0.0 ) ( 00)(.00) ( 50.0 ) A A A A A da B db C dc D dd E de F df da ( )(0.5 ) (.00 4 )(0. ) ( 0.40 )( ) ( 8.00 )(0.06 ) ( )(.0 ²) da.8 ( 50.0 )( 0.0 ) BC A 0.0 ± DE F ( 00 )(.00 ) ( 50.0 ).8 oluto II (repeated ue of earler reult): BC A DE F A A ( 50.0 ± 0.5 )( 0.0 ± 0.) ( 00 ± )(.00 ± 0.06 ) ( 50.0 ±.0 ) ( 50.0 ± % )( 0.0 ± % ) ( 00 ± % )(.00 ± % ) ( 50.0 ±.0 ) ( 50.0 ± % )( 0.0 ± % ) ( 00 ± % ) ( 50.0 ±.0 ) ( 50.0 ± % )( 0.0 ± % ) ( 50 ± 0 ) ( 50.0 ± % )( 0.0 ± % ) ( 00 ± 9.0 ) ( 50.0 ±.0 ) ( 50.0 ± % )( 0.0 ± % ) A A 0.0 ± 9% 0.0 ±.8 ( 50 ± 4% )

8 50 I-8 IGNIFICANT FIGURE AND DECIMAL PLACE Ofte, we wh to quckl etate the preco of our calculated reult wthout applg the rgor of the prevou ecto. To do th, we ue rule that relate the uer of decal place or gfcat fgure we ca keep a calculated value to the uer of decal place or gfcat fgure our data. I. Defto The uer of decal place a uer the uer of dgt to the rght of the decal pot. The uer of gfcat fgure a uer the total uer of dgt, ecluve of leadg zero. II. Eaple The followg tale how the uer of decal place ad the uer of gfcat fgure fve uer. Nuer Nuer of Decal Place Nuer of gfcat Fgure , or 4 The agut the uer of gfcat fgure the lat eaple eal reoved ug cetfc otato three gfcat fgure. III. Rule for Roudg Off Calculated Reult I addto or utracto, keep a a decal place the reult a the allet uer of decal place foud a of the uer eg added or utracted. Eaple: Notce that the uer of gfcat fgure the reult ca e ore tha the uer of gfcat fgure ether uer or le tha the uer of gfcat fgure ether uer.

9 50 I-9 I ultplcato or dvo, keep a a gfcat fgure a the allet uer of gfcat fgure foud a of the uer eg ultpled or dvded. Eaple: 8.75(49.86)/ or 90, ut NOT (.7)(4.9) (.7) (4.9) (4.9) 80 or.8 0 The reaoalee of thee rule a e etalhed ug the ethod dcued the prevou ecto. THE METHOD OF LEAT QUARE Whe two varale, ad, are kow or eleved to have a lear relatohp to each other, the cotat ad of the equato a e otaed fro the eperetal data plottg a graph. The cotat are, repectvel, the lope ad Y tercept of the graph. ujectve judget requred drawg the le that et ft the eperetal data, f the pot are oewhat cattered. The Method of Leat quare allow the calculato of the lope ad tercept for the leat-quare le. The leat-quare le the le that ze the u of the quare of the vertcal dtace of each data pot fro the le. Let u a that we have data pot: (, ), (, ),, (, ). The dagra o the followg page how the le we are ee, alog wth oe tpcal data pot P wth coordate (, ). Let P' e a aocated pot o the le, havg the ae coordate. ce P' le o the le, t coordate. Let e the dtace etwee thee two pot. Let e the u of the quare of all uch dtace. Th the u that we wll ze. 5.5 P P 0 P' 7.5 P'

10 50 I-0 The dtace gve the dfferece coordate of the pot P ad P'. Thu we have ( ). The u the gve ( ) ( ) ( ) a fucto of two depedet varale ad, ad cotat,,,...,,. I uch a cae we ca fd a relatve u (there o relatve au) ettg oth partal dervatve to zero. 0 0 olvg thee equato ultaeoul, we fd ad. The followg a uercal eaple, ug the four data pot (.5,.7), (.87, 6.07), (9.47,.7), ad (.7,.6) u u * The lat row how the u to the correct uer of gfcat fgure. It hould e oted that the uer of gfcat fgure each u creae whe ore data pot

11 50 I- are preet. If there were three te a a data pot, the u of the -coordate would creae aout a factor of. Th would reult a u that would eceed 00, ut would tll e accurate to the earet 0.0. The u would have 5 gfcat fgure. uttutg the u to the equato for ad, ad reeerg that four th cae, we ota ( 6.58) ( 6.58) Our equato for the leat-quare le the, The X tercept, f eeded, a e calculated fro 0.94, ad WARNING: The equato for ad the leat-quare ethod ted to gve zero dvded zero. I order to ota eagful reult, oe eed to have a data pot. MICROOFT EXCEL NOTE: The lope ad tercept of the leat-quare le ca e foud Ecel ug the LOPE ad INTERCEPT fucto. Oe ut e aware of gfcat fgure whe ug thee fucto, ce Ecel doe ot take the to coderato. The ta for the two fucto LOPE(kow_',kow_') INTERCEPT(kow_',kow_') Kow_' the depedet et of oervato or data. Kow_' the depedet et of oervato or data.