Manual:'Conversion'of'Mechanical'Energy'to'Heat'
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1 Manual:'Conversion'of'Mechanical'Energy'to'Heat' Theoreticalbackground Inthiscourseyouhavediscussedthelawofenergyconservation Ui+Ki+W=Uf+Kf.InmanysituationsfrictionalforcesandothernonDconservative forcescanbeignored,howevertherearealsoimportantsituationsinwhichfriction cannotbeignored.frictionalforcesusuallydonegativeworkonasystem,thus removingmechanicalenergyfromthesystem.inthiscontext,mechanicalenergy standsforkinetic+potentialenergy.sincetotalenergyisconservedinnaturethis energycannotjustdisappearbutmustbeconvertedtoanotherformofenergy.the mechanicalenergylostduetofrictionalforcesisturnedintoheat.thegoalofthis experimentistotestthishypothesisandverifythatthemagnitudeoftheworkdone byfrictionisindeedequaltotheamountofheatgenerated. TheworkdonebyaforceFwhichactsonanobjectwhileitmovesadistancedis (1 W = Fdcos Whentheforceisintheoppositedirectionofthemotion(asisusuallythecasefor frictionwehavecosθ=d1sothatthemagnitudeoftheworkisgivenby (2 W = Fd Inordertoobtainasizeableamountofworkwhichonecaneasilymeasureone wouldneedeitheralargeforceoralongdistanceorboth.thisisnoteasytosetup inalaboratorywithlimitedspace.wewillthereforeconsidertheworkdonebya frictionalforcewhichgeneratesatorqueonarotatingcylinder.then,insteadofa largelineardistancewecanrequirealargeangulardistance,i.e.manyrevolutionsof thecylinder.thusweneedtoderivetheequivalentformulato(2forrotational motion. ConsiderapulleyofradiusRwithastringoftensionT& pullingonitasshowninthediagram.ifthepulleyrotatesbyan angleθthentheperimeterofthepulleymovesbyadistanced=rθ. Thustheamountofworkdonebythetensiononthepulleyis (3 W = Td= TR = whereinthesecondstepwehavereplaceddby&rθandinthethirdstepwe introducedthetorqueτ&=&tr&duetothetensionactingatadistancerfromthe rotationaxis.thusthegeneralizationoftheworkformula(2torotationalmotionis& MechanicalEnergyandHeat page1
2 (4 W = Intheexperimentwhichyouwillbeperformingthepulleyisanaluminum cylinderwhichisrotatingonanaxlewithconstantangularvelocity.thestringwill slideoverthesurfaceofthecylinderwithfrictionsothatitgeneratesatorque.this torqueactstoslowdowntherotationofthecylinder.inordertomaintaina constantangularvelocityyouwillapplyacompensatingtorquebyturningacrank whichisattachedtothecylinder. Thefrictionheatsupthecylinderandwewillcomparetheworkin(4with theheatabsorbedbythecylinder.todeterminetheheatyoumeasuretherisein temperatureofthealuminumcylinderandmultiplybyitsmassandthespecificheat ofaluminum (5 Q = c Al m (T f T i IfenergyisconservedyoushouldfindthatW=Qwithintheaccuracyofyour measurements. Apparatus Theapparatusisshown totheright.ameasurable amountofworkisperformed byturningthecrank,which turnsthealuminumcylinder. A nylonropeiswrappedseveral timesaroundthecylinderso that,asthecrankisturned,the frictionbetweentheropeand thecylinderisjustenoughto supportamasshangingfrom theotherendoftherope.this insuresthatthetorqueacting onthecylinderisconstantand measurable.acounterkeeps trackofthenumberofturns. Asthecylinderturns,thefrictionbetweenthecylinderandtheropeconvertsthe workintothermalenergy,whichraisesthetemperatureofthealuminumcylinder.a thermistorisembeddedinthealuminumsothat,bymeasuringtheresistanceofthe thermistor,thetemperatureofthecylindercanbedetermined.bymonitoringthe Aluminum Cylinder with embedded Thermistor Counter Nylon Rope Mass ( 10 kg Crank MechanicalEnergyandHeat page2
3 temperaturechangeofthecylinder,thethermalenergytransferredintothecylinder canbecalculated.finally,theworkperformedandthethermalenergytransferred intothecylindercanbecomparedtodemonstratetheconservationofenergy. Additionalequipmentneeded 1. DigitalOhmmeter'formeasuringtheresistanceofthethermistorinthe aluminumcylinder. 2. Knownmassofapproximately10kg'whichcanbesuspendedfromthenylon rope. 3. Calipers'andascale'formeasuringthemassanddiameterofthealuminum cylinder.thermometerformeasuringtheroomtemperature. Measuringtemperaturewiththethermistor Tomeasurethetemperatureofthealuminumcylinder,athermistorisembedded inside.athermistorisatemperaturedependentresistor.iftheresistanceofthe thermistorisknown,itstemperaturecanbeveryaccuratelyandreliably determined.thecontactsofthethermistorinthecylinderaresolderedtothe copperslipringsonthesideofthecylinder.thebrushesprovideanelectrical connectionbetweentheslipringsandthebananaplugconnectors.bypluggingan ohmmeterintotheseconnectors,theresistanceofthethermistor,andthereforeits temperature,canbemonitored,evenwhenthecylinderisturning. Usetheprovidedgraphofresistanceversustemperatureinthismanualtoconvert yourresistancemeasurementsintotemperaturereadings. MechanicalEnergyandHeat page3
4 Operation ➀Mounttheapparatusonaleveltable.Ifitisnotleveltheropewilltendtoslipand bunchuponthecylinder,whichmakesitdifficulttomaintainasteadytorque. ➁Removethealuminumcylinderbyunscrewingtheblackknobandpullingthe cylinderofftheaxle.measureandrecordm,themassofthealuminumcylinder. Withapairofcalipers,measurethediameterofthealuminumcylinder(attheplace wheretheropewillbewrapped,andrecordtheradius&ofthecylinder.slidethe cylinderbackonthecrankshaft.besurethatthecopperplatedsideisfacingtoward thecrank.alsomakesurethatthepinsonthedriveshaftfitintotheslotsonthe plasticringonthecylinder,thenreplacetheblackknobandtightensecurely. ➂Beforeperformingtheexperiment,spraythesurfaceofthealuminumcylinder lightlywiththeincludeddrypowderedgraphite.&thegraphiteensuresthattherope slidessmoothlyonthecylinder,makingiteasiertoprovideasteady,eventorque, anddecreasingthewearonthealuminumcylinder. ➃ Plugtheleadsoftheohmmeterintothebananaplugconnectors.SetthemultiD metertomeasureresistanceω. ➄Wrapthenylonropeafewturnsaroundthealuminumcylinder.Besurethatthe ropeliesflatagainstthecylinderandhangsdowntheslotprovidedinthebase plate.tieoneendoftheropetothe10kgmass.thereshouldbeonlyenoughturns ofropearoundthecylindersothatthefrictionalpullon theropeisjustenoughtoliftthehangingmassabout'an' inch'off'the'floor'8'no'higher!'toaccomplishthis,wrap theropethreeorfourturnsandcrank.addturnsas neededtosupportthemasswhilecrankingwithonlyvery slighttensiononthefreeend.attachtherubberbandto thefreeendoftherope.now,withoutcrankingandwhile keepingtheropetaughtbytherubberband,tietheother endoftherubberbandtotheeyeboltonthebaseplate.if youfindthatthelargehangingmasscontinuestorise morethan1inchasyouturnthecrank,removeoneturn Cylinder (front view Constant Tension on free end Rubber Band Base Hanging Mass on this end fromthecylindernearestthefreeend.ifthelargehangingmasscontinuestoreston thefloor,addanotherturnofropearoundthecylinderatthefreeend. Youmayfindthattherubberbandisapainintheneck.Therefore,asanalternative, onepersoncanoperatethecrankwhiletheother(beingcarefulnottostandinthe wayputsslighttensiononthefreeendoftherope.byadjustingthetensionand angleofpull,thepersonholdingtheropecanensurethattheturnsoftheropedo notdoubleuponthecylinderandthehangingmassjustliftsoffthegroundandno further. MechanicalEnergyandHeat page4
5 Practiceturningthecrankevenlyandquicklysothatthehangingmassisjustoffthe groundandthecylinderrotateswithconstantangularvelocity.notethatthe cylindergetswarmfromthefriction.donotpracticemorethanaboutadozenturns becauseyouwantthecylindertobenearroomtemperaturewhenyoustartthe actualdatataking.notethatthecounterisincreasingbyonewitheveryturn.atthe endofyourpracticeresetthecountertozerobyturningtheknob. ➅Measuretheresistanceandconvertittotemperatureusingthegraph.Itshould beafewdegreesaboveroomtemperature.notethattheresistanceincreasesasthe cylindercoolsbacktowardsroomtemperature. ➆Youarenowreadyfordatataking.Turnthecrankquicklyandevenlyabout100 times(itisbesttoturnthecrankasquicklyaspossibletoreduceheatlosstothe environment,andrecordtheresultingdifferenceintemperatureofthealuminum cylinder.while'turning'the'crank'pay'attention'to'the'coils'of'the'string'on'the' cylinder,'they'easily'bunch'up'or'overlap'and'raise'the'heavy'mass'to'unsafe' levels.inordertominimizeerrorsduetoradiativeheatlosstotheairyouwantto: 1.determinetheresistanceimmediatelybeforeyoustartcrankingand2.recordthe minimumvalueoftheresistance(maximumofthetemperaturewhichoccursvery soonafteryoustopcranking,usuallywithin1d2seconds.recordthenumbern#of cranksfromthecounter. Repeatstep7threetimes.Notethatthealuminumcylinderneedssometimetocool downbetweenruns.thusyoumightwanttoperformsomedataanalysiswhile waitingforthecylindertocooldown.thecylinderdoesnothavetocoolalltheway downtoroomtemperaturebutthehigheritstemperaturethelargeryourerrordue toradiativeheatloss! MechanicalEnergyandHeat page5
6 Calculations Calculating'W,'the'Work'Performed' Theworkperformedonthecylinderbyturningthecrankequalsτ,thetorqueacting onthecylinder,timesθ,thetotalanglethroughwhichthetorqueacts.itwouldbe difficulttodirectlymeasurethetorquedeliveredbythecrank.however,sincethe angularvelocityofthecylinderismoreorlessconstantthroughouttheexperiment, weknowthatthetorqueprovidedbythecrankmustjustbalancethetorque providedbythefrictionfromtherope.thetorqueprovidedbytheropefrictionis easilycalculated.itisjust: τ=#tr=mgr#wheret#isthetensionintherope,m'isthemasshangingfromtherope, g'istheaccelerationduetogravity,andr'istheradiusofthecylinder.eachtimethe crankisturnedonefullturn,thistorqueisappliedtothecylinderthroughanangle 2π.Thetotalworkperformedthereforeis: W#=#τθ=#MgR#(2πN;' wherem'isthemasshangingfromtherope;g'istheaccelerationduetogravity;r'is theradiusofthealuminumcylinder;andn'isthetotalnumberoftimesthecrank wasturned. ' Calculating'Q,'the'Heat'produced' Theheat(Qproducedbyfrictionagainstthealuminumcylindercanbedetermined fromthemeasuredtemperaturechangethatoccurred.thecalculationis: Q=cAl#m#(Tf#1Ti wherem'isthemassofthealuminumcylinder,cal'isthespecificheatofaluminum (900J/kgK;Tf'isthefinaltemperatureofthecylinder,'andTi'istheinitial temperatureofthecylinder,justbeforecranking. MechanicalEnergyandHeat page6
7 Datasheet' Masshangingfromrope:M= MassofAluminumcylinder:m= Radiusofcylinder:R= Roomtemperature:T= Datafrom3runs: Run Rinitial[kΩ] Rfinal[kΩ] Tinitial[C] Tfinal[C] N Resultsfrom9runs(3ofyoursand6fromotherteams: Run W=MgR(2πN[J] Q=cAlm(TfDTi[J] Ratior=Q/W MechanicalEnergyandHeat page7
8 9 Questions ➀Calculatetheaveragevalue measurements. r oftheenergyconservationratior=q/wforthe9 Calculatethevariancesofthemeasurements: v u s = t 1 n nx (r i r 2 i=1 Arethereanydatapointswhichliesignificantlyoutsidethevarianceinterval?i.e. pointswhicharemorethan2d3timesthevarianceawayfrom r?ifthereare suchpointsdiscussiftheyshouldbediscarded. ➁ Calculatethevarianceofthemean: r v = 1 ux t n (r i r 2 n Isyourmeanvaluefortheenergyratio significantlydifferentfrom1(i.e.isit morethan2timesthevarianceofthemeanawayfrom1? i=1 ➂Discussanysourcesoferrorthatyoufeelmighthaveaffectedyourresults.Are someoftheseavoidable?canyouestimatethemagnitudeoftheeffects? ➃ Isitexperimentallypossiblethattheheatabsorbedbythecylindercouldbe greaterthantheworkperformedonit?explain. MechanicalEnergyandHeat page8
9 Resistance in kilo-ohm Conversion of Mechanical Energy to Heat: resistance of the thermistor versus temperature Temperature in Degree Celsius
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