PPHHYYSSIICCSS 112266

ERROR AND UNCERTAINTY

IInn PPhhyyssiiccss,, lliikkee eevveerryy ootthheerr eexxppeerriimmeennttaall sscciieennccee,, oonnee ccaannnnoott mmaakkee aannyy mmeeaassuurreemmeenntt wwiitthhoouutt hhaavviinngg ssoommee ddeeggrreeee ooff uunncceerrttaaiinnttyy.. IInn rreeppoorrttiinngg tthhee rreessuullttss ooff aann eexxppeerriimmeenntt,, iitt iiss aass eesssseennttiiaall ttoo ggiivvee tthhee uunncceerrttaaiinnttyy,, aass iitt iiss ttoo ggiivvee tthhee bbeesstt--mmeeaassuurreedd vvaalluuee.. TThhuuss iitt iiss nneecceessssaarryy ttoo lleeaarrnn tthhee tteecchhnniiqquueess ffoorr eessttiimmaattiinngg tthhiiss uunncceerrttaaiinnttyy.. AAlltthhoouugghh tthheerree aarree ppoowweerrffuull ffoorrmmaall ttoooollss ffoorr tthhiiss,, ssiimmppllee mmeetthhooddss wwiillll ssuuffffiiccee ffoorr uuss.. TToo llaarrggee eexxtteenntt,, wwee eemmpphhaassiizzee aa ""ccoommmmoonn sseennssee"" aapppprrooaacchh bbaasseedd oonn aasskkiinngg oouurrsseellvveess jjuusstt hhooww mmuucchh aannyy mmeeaassuurreedd qquuaannttiittyy iinn oouurr eexxppeerriimmeennttss ccoouulldd bbee iinn eerrrroorr.. AA ffrreeqquueenntt mmiissccoonncceeppttiioonn iiss tthhaatt tthhee eexxppeerriimmeennttaall eerrrroorr iiss tthhee ddiiffffeerreennccee bbeettwweeeenn oouurr mmeeaassuurreemmeenntt aanndd tthhee aacccceepptteedd ""ooffffiicciiaall"" vvaalluuee.. WWhhaatt wwee mmeeaann bbyy eerrrroorr iiss tthhee eessttiimmaattee ooff tthhee rraannggee ooff vvaalluueess wwiitthhiinn wwhhiicchh tthhee ttrruuee vvaalluuee ooff aa qquuaannttiittyy iiss lliikkeellyy ttoo lliiee.. TThhiiss rraannggee iiss ddeetteerrmmiinneedd ffrroomm wwhhaatt wwee kknnooww aabboouutt oouurr llaabb iinnssttrruummeennttss aanndd mmeetthhooddss.. IItt iiss ccoonnvveennttiioonnaall ttoo cchhoooossee tthhee eerrrroorr rraannggee aass tthhaatt wwhhiicchh wwoouulldd ccoommpprriissee 6688%% ooff tthhee rreessuullttss iiff wwee wweerree ttoo rreeppeeaatt tthhee mmeeaassuurreemmeenntt aa vveerryy llaarrggee nnuummbbeerr ooff ttiimmeess.. IInn ffaacctt,, wwee sseellddoomm mmaakkee tthhee mmaannyy rreeppeeaatteedd mmeeaassuurreemmeennttss,, ssoo tthhee eerrrroorr iiss uussuuaallllyy aann eessttiimmaattee ooff tthhiiss rraannggee.. BBuutt nnoottee tthhaatt tthhee eerrrroorr rraannggee iiss eessttaabblliisshheedd ssoo aass ttoo iinncclluuddee mmoosstt ooff tthhee lliikkeellyy oouuttccoommeess,, bbuutt nnoott aallll.. YYoouu mmiigghhtt tthhiinnkk ooff tthhee pprroocceessss aass aa wwaaggeerr:: ppiicckk tthhee rraannggee ssoo tthhaatt iiff yyoouu bbeett oonn tthhee oouuttccoommee bbeeiinngg wwiitthhiinn yyoouurr eerrrroorr rraannggee,, yyoouu wwiillll bbee rriigghhtt aabboouutt 22//33 ooff tthhee ttiimmee.. IIff yyoouu uunnddeerreessttiimmaattee tthhee eerrrroorr,, yyoouu wwiillll lloossee mmoonneeyy iinn yyoouurr bbeettttiinngg;; iiff yyoouu oovveerreessttiimmaattee iitt,, nnoo oonnee wwiillll ttaakkee yyoouurr bbeett!! EErrrroorr:: IIff wwee ddeennoottee aa qquuaannttiittyy tthhaatt iiss ddeetteerrmmiinneedd iinn aann eexxppeerriimmeenntt aass XX,, wwee ccaann ccaallll tthhee eerrrroorr ΔΔXX.. TThhuuss iiff XX rreepprreesseennttss tthhee lleennggtthh ooff aa bbooookk mmeeaassuurreedd wwiitthh aa mmeetteerr ssttiicckk wwee mmiigghhtt ssaayy tthhee lleennggtthh ll == 2255..11 ±± 00..11 ccmm,, wwhheerree tthhee cceennttrraall vvaalluuee ffoorr tthhee lleennggtthh iiss 2255..11 ccmm aanndd tthhee eerrrroorr,, ΔΔll iiss 00..11 ccmm.. BBootthh cceennttrraall vvaalluuee aanndd eerrrroorr ooff mmeeaassuurreemmeennttss mmuusstt bbee qquuootteedd iinn yyoouurr llaabb wwrriitteeuuppss.. NNoottee tthhaatt iinn tthhiiss eexxaammppllee,, tthhee cceennttrraall vvaalluuee iiss ggiivveenn wwiitthh jjuusstt tthhrreeee ssiiggnniiffiiccaanntt ffiigguurreess.. DDoo nnoott wwrriittee ssiiggnniiffiiccaanntt ffiigguurreess bbeeyyoonndd tthhee ffiirrsstt ddiiggiitt ooff tthhee eerrrroorr oonn tthhee qquuaannttiittyy.. GGiivviinngg mmoorree pprreecciissiioonn ttoo aa vvaalluuee tthhaann tthhiiss iiss mmiisslleeaaddiinngg aanndd iirrrreelleevvaanntt.. AAnn eerrrroorr ssuucchh aass tthhaatt qquuootteedd aabboovvee ffoorr tthhee bbooookk lleennggtthh iiss ccaalllleedd tthhee aabbssoolluuttee eerrrroorr;; iitt hhaass tthhee ssaammee uunniittss aass tthhee qquuaannttiittyy iittsseellff ((ccmm iinn tthhee eexxaammppllee)) ..WWee wwiillll aallssoo eennccoouunntteerr rreellaattiivvee eerrrroorr,, ddeeffiinneedd aass tthhee rraattiioo ooff tthhee eerrrroorr ttoo tthhee cceennttrraall vvaalluuee ooff tthhee qquuaannttiittyy.. TThhuuss tthhee rreellaattiivvee eerrrroorr oonn tthhee bbooookk lleennggtthh iiss ΔΔll//ll == ((00..11//2255..11)) == 00..000044.. TThhee rreellaattiivvee eerrrroorr iiss ddiimmeennssiioonnlleessss,, aanndd sshhoouulldd bbee qquuootteedd wwiitthh aass mmaannyy ssiiggnniiffiiccaanntt ffiigguurreess aass aarree kknnoowwnn ffoorr tthhee aabbssoolluuttee eerrrroorr.. RRaannddoomm EErrrroorr:: RRaannddoomm eerrrroorr ooccccuurrss bbeeccaauussee ooff ssmmaa..ll11 rraannddoomm vvaarriiaattiioonnss iinn tthhee mmeeaassuurreemmeenntt pprroocceessss.. MMeeaassuurriinngg tthhee ttiimmee ooff aa ppeenndduulluumm''ss ppeerriioodd wwiitthh aa ssttooppwwaattcchh wwiillll ggiivvee ddiiffffeerreenntt rreessuullttss iinn rreeppeeaatteedd ttrriiaallss dduuee ttoo ssmmaallll ddiiffffeerreenncceess iinn yyoouurr rreeaaccttiioonn ttiimmee iinn hhiittttiinngg tthhee ssttoopp bbuuttttoonn aass tthhee ppeenndduulluumm rreeaacchheess tthhee eenndd ppooiinntt ooff iittss sswwiinngg.. IIff tthhiiss eerrrroorr iiss rraannddoomm,, tthhee aavveerraaggee ppeerriioodd oovveerr tthhee iinnddiivviidduuaall mmeeaassuurreemmeennttss wwoouulldd ggeett cclloosseerr ttoo tthhee ccoorrrreecctt vvaalluuee aass tthhee nnuummbbeerr ooff ttrriiaallss iiss iinnccrreeaasseedd.. TThhee ccoorrrreecctt rreeppoorrtteedd rreessuulltt wwoouulldd bbee tthhee aavveerraaggee ffoorr oouurr cceennttrraall vvaalluuee aanndd tthhee eerrrroorr ((uussuuaallllyy ttaakkeenn aass tthhee ssttaannddaarrdd ddeevviiaattiioonn ooff tthhee mmeeaassuurreemmeennttss)).. IInn pprraaccttiiccee,, wwee sseellddoomm ttaakkee tthhee ttrroouubbllee ttoo mmaakkee aa vveerryy llaarrggee nnuummbbeerr ooff

mmeeaassuurreemmeennttss ooff aa qquuaannttiittyy iinn tthhiiss llaabb;; aa ssiimmppllee aapppprrooxxiimmaattiioonn iiss ttoo ttaakkee aa ffeeww ((33-- 55)) mmeeaassuurreemmeennttss aanndd ttoo eessttiimmaattee tthhee rraannggee rreeqquuiirreedd ttoo eennccoommppaassss aabboouutt 22//33 ooff tthhee rreessuullttss.. WWee wwoouulldd tthheenn qquuoottee oonnee hhaallff ooff tthhiiss ttoottaall rraannggee ffoorr tthhee eerrrroorr,, ssiinnccee tthhee eerrrroorr iiss ggiivveenn ffoorr ''pplluuss'' oorr ''mmiinnuuss'' vvaarriiaattiioonnss.. IInn tthhee ccaassee tthhaatt wwee oonnllyy hhaavvee oonnee mmeeaassuurreemmeenntt,, eevveenn tthhiiss ssiimmppllee pprroocceedduurree wwoonn''tt wwoorrkk;; iinn tthhiiss ccuuee yyoouu mmuusstt gguueessss tthhee lliikkeellyy vvaarriiaattiioonn ffrroomm tthhee cchhaarraacctteerr ooff yyoouurr mmeeaassuurriinngg eeqquuiippmmeenntt.. FFoorr eexxaammppllee iinn tthhee bbooookk lleennggtthh mmeeaassuurreemmeenntt wwiitthh aa mmeetteerr ssttiicckk mmaarrkkeedd ooffff iinn mmiilllliimmeetteerrss,, yyoouu mmiigghhtt gguueessss tthhaatt tthhee eerrrroorr wwoouulldd bbee aabboouutt tthhee ssiizzee ooff tthhee ssmmaalllleesstt ddiivviissiioonn oonn tthhee mmeetteerr ssttiicckk ((00..11 ccmm)).. SSyysstteemmaattiicc EErrrroorr:: SSoommee ssoouurrcceess ooff uunncceerrttaaiinnttyy aarree nnoott rraannddoomm.. FFoorr eexxaammppllee,, iiff tthhee mmeetteerr ssttiicckk tthhaatt yyoouu uusseedd ttoo mmeeaassuurree tthhee bbooookk wwaass wwaarrppeedd oorr ssttrreettcchheedd,, yyoouu wwoouulldd nneevveerr ggeett aa ggoooodd vvaalluuee wwiitthh tthhaatt iinnssttrruummeenntt.. MMoorree ssuubbttllyy,, tthhee lleennggtthh ooff yyoouurr mmeetteerr ssttiicckk mmiigghhtt vvaarryy wwiitthh tteemmppeerraattuurree aanndd tthhuuss bbee ggoooodd aatt tthhee tteemmppeerraattuurree ffoorr wwhhiicchh iitt wwaass ccaalliibbrraatteedd,, bbuutt nnoott ootthheerrss.. WWhheenn uussiinngg eelleeccttrroonniicc iinnssttrruummeennttss ssuucchh 11..55 vvoollttmmeetteerrss aanndd aammmmeetteerrss,, yyoouu oobbvviioouussllyy rreellyy oonn tthhee pprrooppeerr ccaalliibbrraattiioonn ooff tthheessee ddeevviicceess.. BBuutt iiff tthhee ssttuuddeenntt bbeeffoorree yyoouu ddrrooppppeedd tthhee mmeetteerr,, tthheerree ccoouulldd wweellll bbee aa.. ssyysstteemmaattiicc eerrrroorr.. EEssttiimmaattiinngg ppoossssiibbllee eerrrroorrss dduuee ttoo ssuucchh ssyysstteemmaattiicc eeffffeeccttss rreeaallllyy ddeeppeennddss oonn yyoouurr uunnddeerrssttaannddiinngg ooff yyoouurr aappppaarraattuuss aanndd tthhee sskkiillll yyoouu hhaavvee ddeevveellooppeedd ffoorr tthhiinnkkiinngg aabboouutt ppoossssiibbllee pprroobblleemmss.. FFoorr eexxaammppllee iiff yyoouu ssuussppeecctt aa mmeetteerr mmiigghhtt bbee mmiiss--ccaalliibbrraatteedd,, yyoouu ccoouulldd ccoommppaarree yyoouurr iinnssttrruummeenntt wwiitthh aa ''ssttaannddaarrdd'' mmeetteerr --bbuutt ooff ccoouurrssee yyoouu hhaavvee ttoo tthhiinnkk ooff tthhiiss ppoossssiibbiilliittyy yyoouurrsseellff aanndd ttaakkee tthhee ttrroouubbllee ttoo ddoo tthhee ccoommppaarriissoonn.. IInn tthhiiss ccoouurrssee,, yyoouu sshhoouulldd aatt lleeaasstt ccoonnssiiddeerr ssuucchh ssyysstteemmaattiicc eeffffeeccttss,, bbuutt ffoorr tthhee mmoosstt ppaarrtt yyoouu wwiillll ssiimmppllyy mmaakkee tthhee aassssuummppttiioonn tthhaatt tthhee ssyysstteemmaattiicc eerrrroorrss aarree ssmmaallll.. HHoowweevveerr,, iiff yyoouu ggeett aa vvaa..11uuee ffoorr ssoommee qquuaannttiittyy tthhaatt sseeeemmss rraatthheerr ffaarr ooffff wwhhaatt yyoouu eexxppeecctt,, yyoouu sshhoouulldd tthhiinnkk aabboouutt ssuucchh ppoossssiibbllee ssoouurrcceess mmoorree ccaarreeffuullllyy.. PPrrooppaaggaattiioonn oorr EErrrroorrss:: OOfftteenn iinn tthhee llaabb,, yyoouu nneeeedd ttoo ccoommbbiinnee ttwwoo oorr mmoorree mmeeaassuurreedd qquuaannttiittiieess,, eeaacchh ooff wwhhiicchh hhaass aann eerrrroorr,, ttoo ggeett aa ddeerriivveedd qquuaannttiittyy.. FFoorr eexxaammppllee,, iiff yyoouu wwaanntteedd ttoo kknnooww tthhee ppeerriimmeetteerr ooff aa rreeccttaanngguullaarr ffiieelldd aanndd mmeeaassuurreedd tthhee lleennggtthh ll aanndd wwiiddtthh ww wwiitthh aa ttaappee mmeeaassuurree,, yyoouu wwoouulldd hhaavvee tthheenn ttoo ccaallccuullaattee tthhee ppeerriimmeetteerr,, pp == 22 xx ((ll ++ ww)),, aanndd wwoouulldd nneeeedd ttoo ggeett tthhee eerrrroorr oonn pp ffrroomm tthhee eerrrroorrss yyoouu eessttiimmaatteedd oonn ll aanndd ww,, ΔΔll aanndd ΔΔww.. SSiimmiillaarrllyy,, iiff yyoouu wwaanntteedd ttoo ccaallccuullaattee tthhee aarreeaa ooff tthhee ffiieelldd,, AA == llww,, yyoouu wwoouulldd nneeeedd ttoo kknnooww hhooww ttoo ddoo tthhiiss uussiinngg ΔΔll aanndd ΔΔww.. TThheerree aarree ssiimmppllee rruulleess ffoorr ccaallccuullaattiinngg eerrrroorrss ooff ssuucchh ccoommbbiinneedd,, oorr ddeerriivveedd,, qquuaannttiittiieess.. SSuuppppoossee tthhaatt yyoouu hhaavvee mmaaddee pprriimmaarryy mmeeaassuurreemmeennttss ooff qquuaannttiittiieess AA aanndd BB,, aanndd wwaanntt ttoo ggeett tthhee bbeesstt vvaalluuee aanndd eerrrroorr ffoorr ssoommee ddeerriivveedd qquuaannttiittyy SS.. FFoorr aaddddiittiioonn oorr ssuubbttrraaccttiioonn ooff mmeeaassuurreedd qquuaannttiittiieess:: IIff SS == AA ++ BB,, tthheenn ΔΔSS == ΔΔAA ++ ΔΔBB.. IIff SS == AA-- BB,, tthheenn ΔΔSS == ΔΔAA ++ ΔΔBB ((aallssoo)).. ((AAccttuuaallllyy,, tthheessee rruulleess aarree ssiimmpplliiffiiccaattiioonnss,, ssiinnccee iitt iiss ppoossssiibbllee ffoorr rraannddoomm eerrrroorrss ((eeqquuaallllyy lliikkeellyy ttoo bbee ppoossiittiivvee oorr nneeggaattiivvee)) ttoo ppaarrttllyy ccaanncceell eeaacchh ootthheerr iinn tthhee eerrrroorr ΔΔSS.. AA mmoorree rreeffiinneedd rruullee tthhaatt rreeqquuiirreess aa lliittttllee mmoorree ccaallccuullaattiioonn ffoorr eeiitthheerr SS == AA ±± BB iiss::

TThhiiss ssqquuaarree rroooott oorr tthhee ssuumm ooff ssqquuaarreess iiss ccaalllleedd ''aaddddiittiioonn iinn qquuaaddrraattuurree''..)) FFoorr mmuullttiipplliiccaattiioonn oorr ddiivviissiioonn ooff mmeeaassuurreedd qquuaannttiittiieess:: IIff SS == AA xx BB oorr SS == AA//BB,, tthheenn tthhee ssiimmppllee rruullee iiss ΔΔSS//SS == ((ΔΔAA//AA)) ++ ((ΔΔBB//BB))..

NNoottee tthhaatt ffoorr mmuullttiipplliiccaattiioonn oorr ddiivviissiioonn,, iitt iiss tthhee rreellaattiivvee eerrrroorrss iinn AA aanndd BB wwhhiicchh aarree aaddddeedd ttoo tthhaatt eerrrroorrss iinn AA aanndd BB mmaayy ppaarrttllyy ccoommppeennssaattee::

FFoorr tthhee ccaassee tthhaatt SS == AA22 ((oorr AAnn)),, tthhee eerrrroorr iiss ΔΔSS// SS == 22((ΔΔAA//AA)) ((oorr ΔΔSS//SS== nn((ΔΔAA//AA)))).. AAss aann eexxaammppllee,, ssuuppppoossee yyoouu mmeeaassuurree qquuaannttiittiieess AA,, BB aanndd CC aanndd eessttiimmaattee tthheeiirr eerrrroorrss ΔΔAA,, ΔΔBB aanndd ΔΔCC aanndd ccaallccuullaattee aa qquuaannttiittyy SS == CC ++ ππAA22//BB.. YYoouu sshhoouulldd bbee aabbllee ttoo ddeerriivvee ffrroomm tthhee aabboovvee rruulleess tthhaatt:: ΔΔSS == ((ππAA22//BB))[[22((ΔΔAA//AA)) ++ ((ΔΔBB//BB))]] ++ ΔΔCC ((HHiinntt:: ''ππ'' iiss aa kknnoowwnn ccoonnssttaanntt wwiitthh nnoo eerrrroorr,, aanndd iitt mmaayy bbee hheellppffuull ttoo ddeeccoommppoossee SS aass SS == PP ++ CC,, wwiitthh PP == ππAA22//BB..)) AAss bbeeffoorree,, tthhee mmoorree rreeffiinneedd ccaallccuullaattiioonn rreeppllaacceess tthhee aaddddiittiioonnss iinn tthhiiss ffoorrmmuullaa wwiitthh aaddddiittiioonn iinn qquuaaddrraattuurree.. OObbttaaiinniinngg vvaalluueess ffrroomm ggrraapphhss:: OOfftteenn yyoouu wwiillll bbee aasskkeedd ttoo ggrraapphh rreessuullttss oobbttaaiinneedd iinn tthhee llaabb aanndd ttoo ffiinndd cceerrttaaiinn qquuaannttiittiieess ffrroomm tthhee ssllooppee ooff tthhee ggrraapphh.. AAnn eexxaammppllee wwoouulldd ooccccuurr ffoorr aa mmeeaassuurreemmeenntt ooff tthhee ddiissttaannccee ll ttrraavveelleedd bbyy aann aaiirr gglliiddeerr oonn aa ttrraacckk iinn ttiimmee tt;; ffrroomm tthhee pprriimmaarryy mmeeaassuurreemmeenntt ooff ll aanndd tt,, yyoouu mmiigghhtt pplloott 22ll ((aass yy ccoooorrddiinnaattee)) vvss.. tt22 ((aass xx ccoooorrddiinnaattee)).. IInn ccoonnssttrruuccttiinngg tthhee ggrraapphh,, pplloott aa ppooiinntt aatt tthhee cceennttrraall xx--yy vvaalluuee ffoorr eeaacchh ooff yyoouurr mmeeaassuurreemmeennttss,, SSmmaallll hhoorriizzoonnttaall bbaarrss sshhoouulldd bbee ddrraawwnn wwhhoossee lleennggtthh iiss tthhee aabbssoolluuttee eerrrroorr iinn xx ((hheerree tt22)),, aanndd vveerrttiiccaall bbaarrss wwiitthh lleennggtthh eeqquuaall ttoo tthhee eerrrroorr iinn yy ((hheerree 22ll)).. TThhee ffuullll lleennggtthhss ooff tthhee bbaarrss aarree ttwwiiccee tthhee eerrrroorrss ssiinnccee tthhee mmeeaassuurreemmeenntt mmaayy bbee ooffff iinn aa ppoossiittiivvee oorr nneeggaattiivvee ddiirreeccttiioonn.. TThhee ssllooppee ooff tthhee ggrraapphh wwiillll bbee tthhee gglliiddeerr aacccceelleerraattiioonn ((rreemmeemmbbeerr,, ll==((11//22))aatt22)).. YYoouu ccaann ddeetteerrmmiinnee tthhiiss ssllooppee bbyy ddrraawwiinngg tthhee ssttrraaiigghhtt lliinnee wwhhiicchh ppaasssseess aass cclloossee ttoo ppoossssiibbllee ttoo tthhee cceenntteerr ooff yyoouurr ppooiinnttss;; ssuucchh aa lliinnee iiss nnoott hhoowweevveerr eexxppeecctteedd ttoo ppaassss tthhrroouugghh aallll tthhee ppooiinnttss tthhee bbeesstt yyoouu ccaann eexxppeecctt iiss tthhaatt iitt ppaasssseess tthhrroouugghh tthhee eerrrroorr bbaarrss ooff mmoosstt ((aabboouutt 22//33)) ooff tthhee ppooiinnttss,, IInn ggeettttiinngg tthhee nnuummeerriiccaall vvaalluuee ooff tthhee ssllooppee ooff yyoouurr lliinnee,, yyoouu mmuusstt ooff ccoouurrssee ttaakkee aaccccoouunntt ooff tthhee ssccaallee ((aanndd uunniittss)) ooff yyoouurr xx-- aanndd yy--aaxxeess.. YYoouu ccaann ggeett tthhee eerrrroorr oonn tthhee ssllooppee ((aacccceelleerraattiioonn)) ffrroomm tthhee ggrraapphh aass wweellll.. BBeessiiddeess tthhee lliinnee yyoouu ddrreeww ttoo bbeesstt rreepprreesseenntt yyoouurr ppooiinnttss,, ddrraaww ssttrraaiigghhtt lliinneess wwhhiicchh ppaassss nneeaarr tthhee uuppppeerr oorr lloowweerr eennddss ooff mmoosstt ooff yyoouurr eerrrroorr bbaarrss ((aaggaaiinn,, oonn aavveerraaggee)),, TThheessee mmaaxxiimmuumm aanndd mmiinniimmuumm ssllooppeess wwiillll bbrraacckkeett yyoouurr bbeesstt ssllooppee.. TTaakkee tthhee ddiiffffeerreennccee bbeettwweeeenn mmaaxxiimmuumm aanndd mmiinniimmuumm ssllooppeess aass ttwwiiccee tthhee eerrrroorr oonn yyoouurr ssllooppee ((aacccceelleerraattiioonn)) vvaalluuee..

Fall 2007 PHY126 Experiment 1 FLUID FLOW

Purpose: The purpose of this laboratory is to study some aspects of viscous fluid flow and to see demonstrations of fluid flow and static’s, which can be interpreted by the Bernoulli equation. Part 1 - Viscous Fluid Flow: According to Poiseuille's law, the viscous fluid flow Q (see text) through a tube is pro-portional to the pressure difference P across the tube, so that P = RQ. The resistance to flow is R = 8ηl /πr4 where l is the tube length, r is the radius of the tube, and η is the fluid viscosity. The applicability of this law to water flow through glass capillary tubes is investigated here. The apparatus is very simple. It is sketched below.

The capillary is connected to the bottom of cups A and B. Cup A is positioned a

height h above cup B provided the capillary is horizontal, and that the water in cup A and B is up to the overflow tubes, then the pressure difference on the capillary is just A = ρgh (ρ is the density of water). As water flows from A to B it will collect in the calibrated beaker. By measuring the collection time, the flow rate Q can be computed. It is important to keep adding water to cup A so that the water level stays at the top of the overflow tube.

Procedure:

1. Connect the capillary tube to the cups. Add water to A and pinch the rubber con-necting tubes until the air bubbles are removed - and water commences to flow.

2. You must time the water flow in order to relate the amount of water accumulated in the beaker to flow rate Q. Do this as follows: Put the end of your finger over the hole in cup A so that the water cannot flow through the capillary; empty the calibrated beaker and get it set up; start the timer and remove your finger; add water to cup A as needed to keep the level constant. Do this until a few minutes have elapsed and/or at least 50 ml have collected in the beaker.

State University of New York at Stony Brook

Cup A

h

Cup B

Overflow Tube

Calibrated Beaker Capillary

Overflow Tube

3. Take measurements on two different capillaries with 1 mm and 2 mm diameters. For

each tube, vary h to have at least 4 different values over the range from 5 cm to 30 cm. 4. For each tube, compute R = Δp/Q for each value of h. 5. Now compute R for each tube from the equation given above. 6. Next take the two tubes and hook them first in series and then in parallel, as shown

below. Take new measurements for these combinations. Compute the effective resis tance of the two combinations. You should find R(series) = R1+R2 and R(parallel) = R1R2/(R,I+R2)

Questions: 1. Why is p = ρgh? 2. Within the estimated precision of your experiments, do you find: A) that R. is constant

for a given capillary?, B) that R(exp.) equals R(predicted)?, and C) that R(series) and R(parallel) are given by the above formula? If not, give some possible systematic uncertainties.

Part 2 - Bernoulli Effects, etc.: In this part of the laboratory, you are asked to comment or make computations on the exhibits.

Exhibit I: A container of water has holes iii the sides. Why does the water from hole A not go as far as the water from hole C? Give a quantitative explanation.

Series

R1 R2

R1

R2

Parallel

A

B

C

Exhibit II: There is a Venturi tube apparatus for study. Give a qualitative description and explanation of what you see.

Exhibit III: Hydraulic jacks are very useful for lifting heavy objects by application

of relatively small forces. Study the model jack setup in the laboratory. Push on both sides so that you feel the jack action. What is the principle?

Fall 2007 PHY126 Experiment 2 THERMAL EXPANSION

In this experiment, the coefficients of linear expansion of copper and aluminum will be measured. When a metal rod of length l has its temperature changed by ΔT, its length changes by an amount Δl. For small changes in temperature, the fractional change in length is proportional to the temperature change. The proportionality constant, α, is the coefficient of linear expansion as shown in the equation below:

Δll=α ⋅ΔT

By measuring l, Δl, and ΔT, we will determine the coefficient of linear expansion, α.

A. Equipment

Steam generator, thermometer, battery, voltmeter, “heat tube” with micrometer.

B. Method

A rod of either copper or aluminum rests inside a hollow metal tube (see figure below). The tube and rod start at room temperature. A steam generator can be used to fill the tube with steam and thereby raise the temperature of the rod.

Heat Tube

Thermometersteam steam

Micrometer

Battery

Voltmeter

At each end of the rod is an electrical contact. If the circuit is complete, the voltmeter will read the voltage of the battery. If contact is broken, the voltmeter will read zero. One of the contacts to the rod is made through a small micrometer. By measuring the micrometer settings at which electrical contact is just made for a cool and hot rod, we can measure the change in length of the rod for a known temperature change.

C. Procedure 1. Your apparatus will either contain a copper or aluminum rod. Measure the length of the rod with a meter stick and the starting temperature. 2. Turn the micrometer screw gently back and forth while observing the voltmeter. Find the point at which contact is just barely made and record the micrometer setting. 3. Repeat step #2 several times so that the random error can be estimated. 4. Back the micrometer screw well away from the end of the rod to allow for expansion while heating. 5. Pass steam through the “heat tube” and watch the temperature of the rod as it rises. Wait until the temperature reading has been stable for several minutes. Record the temperature. 6. Using the same procedure as above determine the new position of the end of the rod (and its uncertainty). 7. Calculate the coefficient of linear expansion of the rod. Q1. Does your value of α agree within error with that in the textbook ?

8. Change places with another group and repeat the measurements on the other kind of metal rod. Q2. Compare your result to the accepted value for α.

Fall 2007 PHY126 Experiment 3 THE MECHANICAL EQUIVALENT OF HEAT

Historically, the relationship between heat flow into a material and its resulting temperature change was deduced prior to mankind’s understanding of heat as a form of energy. A unit of heat (the calorie) was invented to quantify heat flow. A calorie is defined as that amount of heat necessary to raise one gram of water by one degree Celsius.

The equivalence of heat energy and mechanical energy was deduced by measuring the amount heat created when an object undergoes a known amount of work due to a non-conservative force. We will use this technique to measure the proportionality constant between the old heat unit (the calorie) and the MKS energy unit (the Joule).

A. Equipment

Mechanical equivalent of heat apparatus, water, stir rod, thermometer.

B. Method

A schematic diagram of the apparatus is shown below. The inner brass cup is partly filled with water. The outer brass cup is connected to a crank handle and turned about the stationary inner cup. The work done by the frictional force is converted into heat.

Thermometer

Brass cup (spins)

Water filled brass cup (stationary)

Since the inner cup is in equilibrium, the torque due to friction is balanced by the torque due to the external force applied to the edge of the aluminum disk. Since work is equal to torque × angle, we find:

W = τ × θ = FR 2πN = 2πR FN (1)

where R is the radius of the disk, F is the force applied to keep the inner cup stationary, and N is the number of turns of the outer cup.

We can analyze the amount of heat input to the system by measuring the change in temperature. In general, Q = mcΔT for a single material which does not undergo a phase change. In our case, several pieces of the apparatus are heated at the same time. The total heat input is:

Q = mwcwΔT + mbcbΔT + mscsΔT + mthcthΔT (2)

where w represents the water, b the brass pieces, s the stir rod, and th the thermometer. Using the known specific heats of each substance (in CALORIES/kg K), we can determine the heat flow in calories and compare this result to the work done (in Joules).

C. Procedure 1. Disassemble the apparatus and measure the mass of each relevant part. The mass of the water can be deduced by weighing the inner cup with and without the water. 2. Reassemble the device with the inner cup 3/4 filled with cold water (roughly 6-8˚ C below room temperature). Stir the water and measure the starting temperature. Place a scrap of paper between the inner and outer cup during reassembly so that the crank will turn smoothly. 3. The crank handle is attached to a counter so that the number of turns can be measured. Record the starting value of the counter and begin to crank. 4. The force necessary to keep the inner cup stationary is read from a spring balance. Record its value. Try to keep the force steady by turning the crank very smoothly and continuously. 5. While one partner cranks, the other should record the temperature of the water. DO NOT Stop cranking the device during these “spot-check” temperature measurements. If fatigue should set in, the lab partners should switch jobs.

6. Continue to crank the apparatus until the final temperature is roughly as far above room temp as the initial T was below it. After cranking is finished, record the new reading on the counter. 7. Take several temperature measurements even after the cranking is finished since the T value will continue to rise for a short time. 8. Calculate the the work in Joules from equation (1) and the heat in calories from equation (2). Determine the ratio, W/Q and compare it to the accepted value. Q1. Why is it best to start the experiment below room temp and end above?

Q2. Why is it necessary to stir the water before each temperature measurement?

Q3. Why is there a residual upward drift in temperature after the cranking is stopped?

Q4. It takes about 100,000 Joules to toast break. Compare this amount to the mechanical work you did during the experiment.

Fall 2007 PHY126 Experiment 4 THE QUEST FOR ABSOLUTE ZERO

The kinetic theory of gases predicts that an ideal gas will obey the relation

pV = nRT (1)

where p is the pressure in Pascals, V is the volume in m3, n is the number of moles of gas, R is the gas constant (8.31J/mol-K), and T is the temperature in K. When V and n are kept constant, we see that equation (1) shows a linear relation between p and T. Using the pressure and temperature characteristics of air, we will estimate the absolute zero of temperature on the Celcius scale.

A. Equipment

Hot plate, beaker, aluminum gas cell, large graduated cylinder with reservoir, pressure transducer, power supply, digital voltmeter, ice water.

B. Method

The pressure transducer is a very clever device using piezoelectric crystals. It outputs an electric voltage proportional to the applied pressure. Piezoelectric crystals produce an electric voltage across their faces when squeezed or stretched. By orienting a number of them carefully inside the transducer, one can measure the applied gas pressure from the produced output voltage. The transducers are calibrated so that they will output 0 volts at 1 atm. The linearity of the transducers will be checked over a small range by changing the height of a column of water pressing on the air in the connecting tube. Next, the tube will be evacuated by attaching it to a vacuum pump, and the voltage output will be compared with the expected value. Finally, the pressure of the gas in the aluminum gas cell will be measured after placing the aluminum gas cell into boiling water and ice water, at temperatures of 100oC and 0oC respectively.

C. Procedure I. Measuring the linearity of the pressure transducer. 1. Attach a pressure transducer to the T-connector at the bottom of the large graduated cylinder and open the clamp. 2. By moving the can up and down, you can change the level of the water in the graduated cylinder. Measure the transducer’s voltage output at 4 different water levels. 3. Using the relation P = P0 + ρgh, calculate the pressure at the transducer for each height of water used in step #2. 4. Graph the voltage of the transducer vs. the pressure of the water. Q1. Is the graph linear? Calculate the slope of the graph.

Q2. What can you say about the linearity of the transducer?

II. Zero pressure 1. Record the output voltage of a dry transducer when it is open to atmosphere. 2. Attach the transducer to the vacuum pump. Open the valve. Wait several minutes for the transducer’s output voltage to stabilize to ensure an accurate result. Record the final voltage in your notebook. 3. Your results from #1 and #2 determine the output voltage of the transducer at 1 atm and 0 atm of pressure. Write a linear equation which expresses the pressure at the transducer as a function of the output voltage (i.e. Pressure = Const1 * Voltage + Const2). This equation represents the calibration of your transducer. III. Pressure thermometry 1. The pressure inside the aluminum gas cell will now be measured at 0oC and 100oC. Attach the pressure transducer to the gas cell and immerse the gas cell in the boiling water. Caution: Boiling water is hot. Proceed with caution! Avoid making any sudden moves that might knock over the apparatus. 2. Record the voltage after waiting a few minutes for the gas to reach the temperature of the water. Calculate, using your calibration equation, the pressure at this temperature. 3. Next, put the gas cell into the bucket of ice water and wait for it to come to equilibrium. Record the voltage. Calculate the pressure.

4. Graph the pressure versus temperature for the points at 100˚C and 0˚C. 5. Extend the line in your graph to pressure = 0. The temperature at this point should be absolute zero. Q3. At what temperature (in ˚C) would the pressure go to zero?

Q4. How does this compare to the expected value of absolute zero?

Fall 2007 PHY126 Experiment 5 SIMPLE HARMONIC MOTION

The phenomenon of simple harmonic motion will be studied for masses on springs and suspended pendulums. Such motion occurs in any system for which the force exerted on a mass is linearly proportional (by a negative constant) to its displacement from equilibrium. The relationship may be written in general as F = -mω2x, where m is the mass, ω is the angular frequency, and x is the displacement from equilibrium.

A. Equipment

1 air track, 1 glider with “flag”, 1 photo-gate and timer, audio tape, small masses, 1 simple pendulum.

B. Method

Throughout the experiment, a system (either a glider between springs, or a pendulum) will be displaced from equilibrium. The period, τ, of the resulting motion will be measured. The dependence of the period on the amplitude of the motion will be assessed. For true simple harmonic motion, there should be no dependence of the period on the amplitude.

C. Procedure I. Measurement of the Spring Constant. 1. Your glider is held between two springs. Record its equilibrium position. 2. Attach a piece of audio tape to the glider and lay it across the “air pulley” with a small mass suspended on the end of the tape. 3. Measure the displacement of the glider from equilibrium for 4 different hanging masses. 4. Graph the weight of the hanging mass (y axis) vs. the measured displacement. Q1. Is your weight-displacement curve linear? Comment.

5. Determine the spring constant, k, from your graph.

II. Measurement of the Period of the Motion. 1. Remove the audio tape and place the photo-gate so that it is centered over the flag on the glider when the glider is at equilibrium. Measure the mass of the glider. 2. Set the Pasco Timer to “two-pulse” mode. This will measure the time interval between two successive times at which the flag blocks the photo-gate. 3. Start the system oscillating. Measure the time between successive blockages of the photo-gate using the Pasco timer. This time is 1/2 of the system period. Q2. Present an argument proving that the timer measures only 1/2 of the system

period.

4. Measure the 1/2 period several times using the “reset” button on the timer. 5. Compare your results to the theoretical value τ = √(m/k). 6. Repeat the measurement for 2 other masses by taping extra mass to your glider. III. Maximum velocity and position. 1. Measure and record the width of the flag atop your glider. 2. Set the Pasco Timer to pulse width mode. In this mode the timer will measure the time interval during which the light is blocked. 3. Start the oscillator by pulling it back a measured distance from equilibrium and releasing. 4. Record the time interval for which the gate is blocked and calculate from this the glider’s velocity as it passes through equilibrium. 5. Check conservation of energy by calculating both 1/2kA2 (maximum potential energy) and 1/2 mvmax

2 (maximum kinetic energy) Q3. Do the energies measured in part 5 agree within error?

6. Repeat the measurement for several amplitudes. IV. Simple Pendulum. 1. Measure and record the length of the simple pendulum. 2. Place the timer in two pulse mode. Operate the timer manually and measure the time required for 20 swings. Repeat several times.

3. Compare the measured period to the theoretical value. Repeat for a different length of the pendulum. 4. Measure the period as a function of amplitude of the swing. Q4. It it independent? Why or why not?

Fall 2007 PHY126 Experiment 6 STANDING WAVES

In this experiment, standing waves will be observed in a vibrating string. The wavelengths of the waves and the tension in the string will be measured. From these measurements and the frequency of the wave, the mass per unit length of the string will be determined.

The velocity of transverse traveling waves in a stretched string is given by

v = Tμ

where T is the tension in the string and μ is the mass per unit length of the string. The wavelength of the wave is then

λ =vf

=1f

Tμ

where f is the frequency of the wave.

In your apparatus, a 60 Hz traveling wave is generated by a vibrating reed attached to one end of the string. The tension in the string is generated by a weight hung from the other end of the string. (See Fig. 1) The traveling wave is reflected back along the string at the pulley, and this reflected wave is reflected again at the vibrator. When the second reflected wave is in phase with the original wave, a standing wave pattern will be observed in the string. The distance between nodes is one-half the wavelength of the wave.

1. Place a 100 g mass on the end of the string. Adjust the position of the vibrating reed until a stable, standing wave pattern is observed. Measure the wavelength λ of the wave, and estimate the uncertainty.

2. Try to make standing wave patterns for several different positions of the reed (each different by 1/2 wavelength).

3. Repeat the above procedures using at least three other masses

Plot λ2 as a function of T, the tension of the string, and from the slope of this curve (and your knowlege of the frequency!), determine μ, the mass per unit length of the string. Estimate the uncertainty in this result. Compare with the mass per unit length determined by weighing a known length of string.

4. Repeat the procedure using a piece of copper wire in place of the string. Q1. What do you expect to change?

Fall 2007 PHY126 Experiment 7 THE VELOCITY OF SOUND

In this experiment we will determine the velocity of sound in air by studying the propagation of 40 kHz sound waves between two piezoelectric crystal transducers, one acting as a transmitter, the other as a receiver. An oscilloscope will be used to measure the time difference between the sending of a short sound pulse and its subsequent receipt at a microphone a known distance away.

A. Equipment

1 Dual Trace Oscilloscope, 1 Function Generator, 2 Piezoelectric Crystal Transducers, 1 Meter Stick, 1 DC Power Supply, 1 Frequency Counter.

B. Method Sound waves are created by local variations in pressure above and below the ambient. These waves travel through the air at swift but finite speed. We will measure the speed of sound waves in air by measuring the time required for a short sound pulse or “chirp” to travel from its source to a receiver. This time interval is short and is measured with the help of an instrument called an oscilloscope. The experimental setup is shown in the figure below:

Oscilloscope

Ch 1 Ch 2

Function Generator

The function generator will be set to produce a square pulse. Each edge of the square pulse will chirp the piezoelectric transmitter crystal. The function generator is wired to both the crystal and to the oscilloscope. The sound is received at a second crystal. A small amplifier is mounted to the receiver to make its signal large enough to be easily visible on the ‘scope.

The oscilloscope is essentially a voltmeter in which a spot on a screen can be displaced up or down or left or right by an amount proportional to an applied voltage. The spot can also be made to move from left to right at a uniform rate (known velocity) to show the time variation of a voltage that deflects the spot up and down. The latter of these two modes is used in our experiment. The oscilloscope sweep is triggered (started) by the function generator when it chirps the transmitter crystal. The screen then shows two traces, one for the transmitted pulse (always at the left edge), and one for the received pulse (delayed). By knowing the velocity of the trace, we can measure the delay of the received signal and hence the velocity of sound.

The oscilloscopes used here are expensive high-quality instruments and should be handled carefully. Do not have the intensity control set too high, and never allow a small very bright spot to remain stationary on the cathode-ray tube face. The screen can easily be damaged if the intensity is too high.

.

C. Procedure 1. Set the function generator to produce maximum amplitude square waves at a frequency of 100 Hz. 2. The oscilloscope should be set to trigger from channel 1 in the AUTO mode and the VERT MODE switch should be set to DUAL, so that two traces are visible. Your TA will lend assistance if the oscilloscope is adjusted improperly. 3. The receiver transducer has a small amplifier built onto its holder because the received signal is quite weak. Adjust the 'scope setting until you can clearly observe the elapsed time between the edge of the square wave and the arrival of the sound pulse at the receiver. 4. Move the receiver back and forth and observe the motion of the received pulse on the screen. If the pulse appears to move in proportion to

the distance between the transmitter and receiver, then your device is adjusted properly. Q1. Sketch both traces and explain clearly what is happening.

5. Record the setting of the “time base” of the oscilloscope. This setting tells you the inverse of the velocity of the trace. For example, if your time base setting were 10.0 milliseconds per centimeter, then a spacing of 3.6 centimeters between pulses would imply a 36 millisecond delay. 6. Record the location of both the transmitter and receiver. Record the distance (in cm) between the transmitted pulse trace and the received pulse trace on the oscilloscope screen. 7. Calculate the time delay of the received pulse using the oscilloscope time base setting. 7. Repeat steps 5-7 for five additional spacings of the transmitter and receiver. 8. Plot the distance of separation (vertical axis) vs. the time delay. Calculate the velocity of sound from your graph. Q2. Why did we not just use the distance between the transducers for this part?

(HINT: What is the exact position of each transducer?)

Q3. Compare your values with the theoretical value:

csound =γ air RT

Mair

Assume that air is 78% N2, 21% O2, and 1% Ar. A table of atomic weights can be found in Appendix D of your text and values of γ are listed pp. 495.

Fall 2007 PHY126 Experiment 8 LENS OPTICS

In this experiment we will investigate the image-forming properties of lenses, using the thin-lens equation:

1s+

1s'=

1f

where s and s’ are the object and image distances from the lens and f is the focal length of the lens.

A. Equipment

1 Optical Bench, Several Convex and Concave lenses, 1 Light source, 1 Mirror, 1 Screen.

C. Procedure

I. Measuring Focal Length 1. The focal length of a lens is equal to the image distance of an object at infinity. Image the overhead lights on a piece of paper and measure the distance from the paper to the lens. Q1. Prove using the thin lens equation the statement from step 1.

2. Set up on your optic bench the equipment shown below:

Mirror

Object

Image

Lens

3.Move the lens until the reflected image forms a clear focus at the same position as the object. Determine how to get the focal length from this configuration.

Q2. Is the location of the mirror important?

Q3. Draw a ray diagram to show how the image is formed.

4. Repeat for each of your lenses.

II. Converging lenses and Real Images. 1. For each of your convex lenses, measure the s and s’ (as well as the image size) for at least six different object distances. Q4. What is the smallest object distance for which a real image can be formed?

2.For each lens, make a plot of 1/s’ cs. 1/s. Determine the focal length of each lens from the plot and compare with your previous results. 3. Use your measurements to test the relation m = -s’/s. III. Two Lens Systems. 1. Mount a diverging lens on the optical bench in front of an illuminated object. Q5. Where is the image formed? Can you see it? Why?

2. Put a strong (short focal length) lens behind the concave lens. If the converging lens is strong enough, it will be possible to form a real image. 3. The virtual image created by the diverging lens acts as the object for the converging lens. Using the measured distance to the image of the converging lens (s’ for that lens), and its known focal length, measure the location of the virtual image from the diverging lens. 4. Using the known object location and JUST MEASURED image location, calculate the focal length of the diverging lens. 5. Construct a simple two-lens telescope. Try a few different lens combinations to verify the relationship between the focal lengths of the lenses and the magnifying power.