Using Potentil Enegy You ve job poviing te engineeing elp o n citect in Coloo. You e cuently esigning cble tow to pull sies up ill so tey cn si own. e custoe woul lie te cble tow to pull sie upill t constnt cceletion o te botto ecing spee o 6 /s t te top. You nee to eteine wt type o cble you soul pucse. e ill is 5 long n incline t egees o te oizontl. By esuing sie spees on ownill un, you now tee is iction oce o 5 between te sis n te snow inepenent o te sie s weigt. FOCUS v o = F =5 Question: t is te oce on te cble? Appoc: o 5 v = 6 /s Use consevtion o enegy to elte te inl spee n te oces. Inclue te Et in te syste n use gvittionl potentil enegy Initil tie: just te gbs te ope t botto Finl tie: just s sie gets to te top e ope oce (), n te ictionl oce () ve coponents long te isplceent. Enegy tnse occus Coose one xis long te si slope Migt nee to estite ss o sie PHYSICS DESCRIPIO Fee boy ig y i Enegy ig vi = xi Foce ig Syste: sie + Et x get untity: oizontl initil stte Enegy tnse inl stte y i = = v = 6 /s vi = = 5 = 5 Geoety v Quntittive eltionsips: Consevtion o Enegy E - Ei = Einput - Eoutput Syste enegy is inetic + potentil E i = KE i + GPE i = + E = KE + GPE = v + g Enegy tnsee to n o syste Einput = Eoutput = l = l = l = l = l = l = Consevtion o Enegy v +g- = - Pln: Fin Consevtion o enegy v +g- = - Fin sin = 3, eutions v + gsin = - v +gsin + = ee to estite te sie s ss i it oes not cncel out. Mss o sie oes not cncel Estite it t g, v +gsin + = Cec units [] = [oce], [] = [oce], [g sin ] = [oce] [ v ] = [ss] [ / s ] [] = [ss] [cceletion] = [oce] o ll units e oce units = ( 5 ) ( g) (6 / s ) + ( g) (9.8 / s ) sin + (5 ) 36 + 335 + 5 = = 5 Se s beoe
A ien s cil is plying wit toy cs n you ecie to elp by builing loop-te-loop tc. You stt c on te enty tc bove te igest point on te cicul pt o te tc. e c goes own te enty tc oun te cicle n up n exit tc. Bse on te stting eigt o te c, you ecie to clculte te spee o te c wee it entes te cicul pt o te tc s well s t te top o te cicul pt o te tc n on te exit tc c bove wee you stte. v = c Exple v c c b v b Use consevtion o enegy. syste: object + Et E i = KE i +PE i Fo to b E i = g E = v b E - E i = DE tnse DE tnse = v b - g = v b = E = KE +PE g Assue iction n i esistnce not ipotnt Initil tie t top o entnce p. Assue te c stts o est. Finl tie t botto o cicle. Fo to c E i = g E = v c + g c v c + g c - g = v c = g - g c Fo to E i = g E = v + g v + g - g = v v = v c = g - c = g - g ( ) g( - ) > not possible Mxiu eigt wen KE = All enegy is PE Mxiu eigt is initil eigt i initil KE = o enegy input Exple You copny s been ie to esign stunt o new ice sow. e st o te sow entes by iing on sle wic stts o est t te top o cuve ice tc bove te suce o te ice in. e tc les own to te in n, t tt point, becoes veticl cicle wic etuns gin to te in. You job is to clculte te xiu ius o te cicle so tt tis ing loop-te-loop cn be one witout injuing te ig pice st. Assue tt you cn neglect iction n i esistnce s ist ppoxition. v o = v t t is te lgest ius suc tt cicul otion is possible? = v Mxiu ius ens iniu cceletion Acceletion cnnot be slle tn g t top o cicle Get cceletion o oces on sle. Get velocito consevtion o enegy Syste: sle + Et Gvittionl potentil enegy Initil tie: stt t top o tc. Finl tie: t top o cicle Ignoe iction, i esistnce v Fee-boy Dig o sle t top o cicle F G F y = y iniu cceletion ens iniu oce Initil Enegy v o = = = v o = =? F y = -F G - F y (in iu ) = F G -g= - (iniu) = g Finl Enegy y t v t y t = v t =? E i = KE i +PE i = g E = KE + PE = v t + gyt
Consevtion o Enegy: E - E i = E in - E out v t +gy t -g = get: Fin v t + g( )- g = v t +g Fin v t = v t Fin g = ( ) - g = v t 3 3, 3 eutions o [3] into [] g = v t g = v t g + g ( )-g = + ( ) - = 5 = = 8 Into [] units e o bot sies istnce units e xiu eigt o te sle wen tvelling oun te cicle is 6, less tn te initil eigt o. is is esonble since soe o te initil potentil enegy s becoe inetic enegy t te top o te cicle. Exple A sie stts o est on te slope on suit n ten sis ove two successively lowe ills o eigt n. e lowest ill is essentilly sei-cicle centee t eigt. e sie wnts to leve te lowest ill t its top n ly toug te i n ss you ow up te slope to stt gliing own te ill. Assue iction n i esistnce e negligible. v o = t is te initil eigt o te sie to leve te secon ill t its top? Sie stys on ill i te oces on te sie give te cceletion necessy to go in cicle. = v Get te spee on te top o te ill o consevtion o enegy. Syste: sie + Et Initil tie: stt on slope Finl tie : top o n ill Get te necessy cceletion o oces v Fee-boy Dig o sie on top o lst ill I sie leves te ill = - g = - Initil Enegy v o = - = - Fo te sie to ollow te cicul ill =? v o = = =? E i = KE i +PE i = g DE tnse = = v Finl Enegy v v =? E = KE + PE = v Consevtion o Enegy E - E i = DE tnse v -g = tget: Fin v - g = Fin v = v Fin g = 3, 3 eutions [3] into [] [] [3] [] g = v g = v g - g = = into [] v coect units bot sies e istnces = 5 bove te top o te ill But tt will not get you ove te st ill ee to be t eigt o t lest up te slope
Exple Fo you lbotoy expeience, you now it is iicult to esue te coeicient o inetic iction between two suces. One o you lb ptnes suggest using sping to popel bloc up p incline t n ngle o te oizontl tt you esue. e bloc is to be el ginst sping, copessing te sping istnce o its elxe position tt you esue. en te bloc is elese, te sping expns n puses te bloc upw long te p. e bloc leves te sping, going istnce up te incline tt you lso esue? You cn lso esue te ss o te bloc n te sping constnt. ill tis poceue give you wt you wnt? v o = v = t is te coeicient o inetic iction? Fee boy ig µ Use consevtion o enegy: syste: bloc, Et, n sping Potentil Enegy: sping, gvittionl Enegy tnse: iction Initil tie: just te elese o copesse sping Finl tie: just wen bloc stops Get ictionl oce o ynics - y = y = -g cos = +z z o +z z z o Initil ie Finl ie v = x o x s E i = KE i + PE i = x E = KE + PE = g Consevtion o Enegy: tget: v o = Enegy nse v x o E out = x g - = - x g - = - x E - E i = E in - E out Fin = Fin -g cos = Fin g - = - Fin sin = Geoety Pln [4] [] [] [3], 4, 4 eutions Execute te pln o te botto up. Put [4] into [3] n solve o g sin - = - - g sin + = Put into [] long wit [] n solve o - g sin + = g cos -g sin + = g cos cec units [ oce ] + oce is tn ce [ is tn ce] [ ] [ oce ] Coect, s no units = [ ] How to Solve Pobles Using Enegy. Pictue te sitution t is te syste? How is it oving? Is tee enegy tnse? t pt oes te object tvel? Is tee potentil enegy? Ceully ientiy te initil tie n te inl tie you wnt to consie. Cn you ccount o ll o te enegy o you syste t tose ties? KE + PE Cn you ccount o ll enegy tnses between tose ties? l E = tnse F l l o
. Deine you untities wit espect to coointe syste. Me sue you now wic iection is + n wic is -. Foce, position Use you eine untities to wite own te consevtion o enegy eution o you syste. Keep tc o te signs. Keep tc o te tget untity. Do you nee to now nyting else in ition to consevtion o enegy? 3. Ientiy ll in you consevtion o enegy eution n elte te by eutions to ote inotion o pinciples pysics. 4. Solve te syste o eutions to get you tget untity. 5. Cec you nswe Coect units? Resonble bevio o vlue? Di you nswe te uestion? Foce lws Kinetics