11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

Transcription

1 11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure P agains ime should look like. 400 P I hrow his quesion ou o he class and hey draw a graph like ha a he righ. [They learned heir waer-ank lesson well.] As air flows ou, P decreases, bu as he pressure in he ire ges less, here s less force pushing he air ou and so he air will flow ou more slowly, so herefore he pressure will go down more slowly. Tha seems o make he graph concaveup. Evenually he pressure drops o zero and he graph his he -axis. The problem I give hem now is o consruc a model for he P- relaionship. Tha is, you are required o use wha undersanding of air pressure you already have, or can acquire, o deermine exacly how he pressure in he leaking ire should change over ime, and hence find he form of an equaion relaing P o. Le s sar wih a conjecure. Wha sor of equaion do you hink i migh be? Isn i a parabola? Why would you hink ha? Remember he waer ank experimen? We go a parabola ou of ha, and isn his really he same hing? Is i? Is a ire he same as a waer ank? Grea nosalgia for me when hey ask his quesion because I am reminded of a cerain Sunday afernoon, long ago, when I was young and green and hinking abou his very quesion, he waer ank and he ire boh wih a small hole, how hey were he same and how hey were differen. And a one poin I convinced myself ha hey should be prey much he same and ha he air pressure graph should herefore a parabola. 0 Relaive and absolue pressure When you pu your pressure gauge on he ire you read 400 kpa and you hink ha s he pressure inside bu you re wrong. I s acually 500 kpa. The gauge measures pressure relaive o he ouside air, and ambien air pressure is 100 kpa. If you ook he same ire ino ouer space, he gauge would read 500 kpa. Here we always work wih relaive pressure. Tha s wha everybody always does. And ha s all very well, bu he rouble was ha I had anoher compleely reasonable argumen ha i wasn a parabola, an argumen ha was prey imporan o me because i was based on my concepion of wha air really was, a whole collecion of iny randomly moving molecules. So was I abou o lose his long-held concepion? I was in quie a quandary and had o call up one of my physics profs o ge my hinking sraigh. Read on! 11. ire pressure 1

2 Is a ire he same as a waer ank? Le s sar wih he waer ank. The waer ges pushed ou of he hole because of he pressure, and ha s deermined by he amoun of he waer in he ank, and as he waer flows ou he pressure ges less so he flow rae decreases. Now ha s wha happens in he ire as well. The air ges pushed ou because of he pressure, and ha s deermined by he amoun of air in he ire, and as he air flows ou he pressure ges less so he flow rae decreases. There seems o be a big similariy here. Bu i urns ou ha waer and air are differen in an essenial way. Waer pours ou of a hole in he ank in quie a differen way from he rush of air ou of a hole in he ire. And he difference, in a word, is graviy! The main facor behind he flow of waer ou of he ank is he force of graviy. For example i makes a difference o he flow rae wheher he hole is a he boom of he ank or half-way up. Bu for he ire, he force of graviy is negligible i doesn maer wheher he hole is a he op or he boom of he ire he flow rae will be he same. Le s ge down o basics. The air in he ire consiss of a large number of molecules, which are consanly in moion. Now wha happens when one of hose molecules his he inside surface of he ire? Well, i bounces off. And in fac ha s wha causes he pressure in he ire. When you poke he ire and push i in, why does i pop back ou when you ake your finger off? because all hose molecules are colliding wih he inside surface of he ire and pushing i back. Now. Wha happens if here s a hole so ha a molecule heading for he inside surface of he ire his he hole insead? Well i shoos ou and escapes. And ha s wha causes he flow ou of he hole. Surprisingly enough, ha s all here is o i. Every molecule ha flows ou is one ha was heading for he surface of he ire, minding is own molecular business, and found a hole insead. I lead my sudens hrough his analysis wih as lile promping as possible. In fac I am impressed by how much hey seem o be able o do, a leas he ones who are prepared o conribue. The sudens have a endency o hink of he ire as a balloon, wih he air being pushed ou because of he elasic force of he ube as i conracs. Bu when he ube is imprisoned inside he ire, i doesn srech as i ges filled. If i did, i would be weaker. So i s no like a balloon a all. In fac i s beer o hink of he ire as a rigid srucure made of hard plasic wih air pressure inside. Pu your finger jus above he hole and feel ha lile je of air. Those molecules are jus escaping by chance? Wow. Jus o emphasize he difference, suppose we ook he waer ank ou ino space where here was no graviy. Then here d be nohing o sop he waer molecules from wandering around he inside of he ank and we d have o pu a lid on he ank or hey d wander ou he op. Bu wha abou he hole? Well hey d wander ou of ha oo, in jus he same way ha he air molecules wandered ou of he hole in he ire. And in his case, he z-curve of he amoun of waer remaining in he ank wouldn be a parabola anymore, i would be he same kind of curve as we ge for he ire. 11. ire pressure 2

3 So wha is he air pressure curve anyway? I s ime o ask jus wha he ire curve migh be. Consider he following quesion. Suppose your ire has a small leak. A one poin you measure he pressure o be 400 kpa. Suppose over he nex minue i drops o 384 kpa a loss of 16 kpa. Now you leave i for a while unil i s dropped o 200 kpa. Half of wha i sared wih. So here s he quesion How much will i lose in one minue now? I hrow his problem ou o he class and am impressed by how many of hem ge i righ and even seem o have a good feeling for he reason. Tha s he nice hing abou molecules hey re quie inuiive. Well, here s a simple argumen. We need o hink in erms of our model he reason ha molecules escape from he hole is ha hey happen o encouner i in heir random moion. When he pressure is 200, here s half as many molecules in he ire as here were when i was 400, so here will be half as many collisions wih he hole, so he flow rae ou should be cu in half. So insead of losing 16 kpa in he nex minue, i loses 8. And so forh. When i has dropped o 100 kpa, i will lose 4 kpa in he nex minue. Wha his argumen is really saying is ha he amoun los in a minue ough o be always in proporion o he amoun in he ire a he sar of he minue. A formula for P. Well, ha s a beauifully simple argumen, and we can even ge an equaion ou of i. Le s firs formulae his air loss principle precisely. Wha have we said? Tha he pressure loss over a one minue inerval is proporional o he pressure a he sar of he minue. Alernaively saed: every minue he ire pressure drops by a fixed percenage. Wha is his percenage for our hypoheical example? Well, when P = 400, he loss is 16, and when P = 200, he loss is 8, and when P = 100, he loss is 4, so he 1-minue loss is always 4% of he saring pressure. Thus: every minue he ire pressure drops by 4%. Does his give us a formula? Yes i does. Think in erms of he muliplier. A 4% loss is he same as muliplicaion by 96%, so ha every minue he pressure muliplied by 0.96 If he pressure sars a 400, hen afer minues i will be: P() = 400(0.96). We have our P-funcion! Our pressure loss curve is no a parabola i s an exponenial decay funcion. Parabola and exponenial An ineresing ake on he difference beween hese wo modes of decrease is o ask how he flow rae ou of he hole depends on he amoun in he vessel. For he ire (exponenial decay), as we now see, he flow rae ou is proporional o he pressure iself. Wha abou he waer ank (parabola)? In his case he flow rae ou is proporional o he square roo of he waer deph. In he firs case, he flow rae will be cu in half when he pressure is halved. In he second case, o cu he flow rae in half we have o cu he amoun of waer o ¼. 11. ire pressure 3

4 Experimen. The quesion we are sudying is how he pressure P in your ire goes down if here s a small leak, and in he las secion we developed an exponenial decay model. I s ime o perform an experimen o check his ou. In fac i s no so easy o work wih a ire. The problem is ha he very ac of measuring he pressure alers is value (hiss!) and for somehing as small as a ire his can be significan. So we used an old car ire insead which we go from a garage. We pumped i up o 400 kpa (ha s a very hard car ire hey are normally jus over 200) and one of he sudens drilled a hole ino he side wih a small drill, and i worked perfecly. Wih a large ire, he amoun of air los in he ac of measuring he pressure is negligible relaive o he amoun in he ire. In fac he pump we used had an in-line gauge and all we had o do was monior ha. We used he ire in anoher class a couple of days laer and he hole had o be poked, bu i worked fine again. The ire now sis in a safe place in he school and is a resource for all ime. Ever school should have an old car ire. We ook readings every 5 minues for an hour. The daa are recorded below and ploed a he righ. The gauge was such ha one could ake a reading accurae o 5 kpa. ime (min) pressure P (kpa) Who wans he drill? I held he elecric drill up in he air and asked who waned o drill he hole. We had expeced a grea rush o he fron bu here was sillness. Hasn everybody always waned o drill a hole in a car ire? Perhaps hey were jus awed by he prospec. Of course afer a few momens here was a graifying response P The graph cerainly looks reasonable. I has he concave-up shape ha our heoreical model prediced. Now wha do we wan o do wih i? We wan o es wheher i has he exponenial form. 11. ire pressure 4

5 Tesing he exponenial model Now we es wheher our daa has he exponenial form: P = ac. So how do we do ha, and how do we find he righ values of a and c? A sandard calculaion shows ha P will be an exponenial funcion of ime if he logp graph is a sraigh line. So we abulae and plo logp. This is done below. And holy cow he poins do lie in a quie a wonderful sraigh line. We can conclude ha he P- daa is exponenial. Of course, his log ransform resul is no somehing you need o remember. Beer o reconsruc i each ime by hiing he arge equaion wih he log funcion. P = ac logp = log(ac ) logp = (loga) + log(c ) logp = (loga) + log(c). This ells us ha logp is linear in. P logp logp We fi a rend-line and ge he equaion: logp = Now o ge he P-equaion we exponeniae wih base 10: P = = = ( ) = 385 (0.974) Noe ha he saring value (=0) ha he formula gives us is p=385. Tha suggess ha a he beginning we didn ge he ire quie up o 400. And wha is he percenage loss ha our formula predics? In each minue he pressure is muliplied by a 2.6% loss per minue logp Acually ha saring value of 385 is no acually such a big error from our measured 400. The relaive error, 15/385, is only 4%, and if you abulae he exponenial formula you ll find ha some of he oher poins have almos he same inaccuracy.] 11. ire pressure 5

6 Problems 1. A he momen (=0) my ire has pressure P = 300 kpa, bu i has a slow leak and loses 5% of is pressure every hour. (a) Wha will is pressure be afer en hours (=10)? [Answer: ] (b) Find a formula for P afer hours. [Answer: P = 300(0.95).] (c) A wha ime will P = 150 kpa? [Answer: =13.5 h.] 2. A he momen (=0) my ire has pressure 360 kpa, bu i has a slow leak and loses 2% of is pressure every hour. (a) Wha will is pressure be afer one hour (=1)? (b) Wha will is pressure be afer five hours (=5)? (c) Find a formula for is pressure P afer hours. (d) Wha is he percenage loss in pressure over a en hour period? (e) How long unil he pressure drops o 10 kpa? 3. My bike ire has pressure 360 kpa, bu afer a week his has fallen o 320 kpa. (a) Wha will i be afer anoher week? (b) Find a formula for he pressure P afer weeks. (c) I can be bohered o fix he ire and I refuse o ride i when i s below 100 kpa and I don have a pump and I can only go o he gas saion once a week on Saurday mornings, and you have o pay 25 cens for air, so I only fill i up when I have o. Assuming ha each pump-up brings i up o 360 kpa, how long is he inerval beween pump-ups? 4. The daa on he lef was colleced from a leaking ire. Plo logp agains and deermine ha his daa lies in a reasonable sraigh line. Draw he bes-fi line, eiher wih eyeball and ruler or wih a regression rouine, and find is equaion and ransform his equaion ino an exponenial equaion for P. 5. I measure my car ire pressure a noon, a 1 PM and a 2 PM. During he firs hour i los 30 kpa and during he second hour i los 26 kpa. Wha was he pressure a noon? 6. I measure my bike ire pressure a noon, a 1 PM a 2 PM and a 3 PM. During he second hour i los 80% of wha i los during he firs hour and during he hird hour i los 50 kpa. Wha was he pressure a noon? P ire pressure 6