Two-liquid Cartesian diver

Transcription

1 SPECIAL FEATURE: FOOD PHYSICS Two-liquid Cartesian diver G Planinšič 1,2,MKos 2 and R Jerman 2 1 Deartment of Pysics, Faculty for Matematics and Pysics, University of Ljubljana, Slovenia 2 Te House of Exeriments, Slovenian Hands-on science centre, Ljubljana, Slovenia Abstract It is quite easy to make a version of te well known Cartesian diver exeriment tat uses two immiscible liquids. Tis allows students to test teir knowledge of density and ressure in exlaining te diver s beaviour. Construction details are resented ere togeter wit a matematical model to exlain te observations. M Tis article features online multimedia enancements Te Cartesian diver is one of te most oular simle exeriments. It is always attractive even if you ave seen it many times before. In addition, tis simle exeriment offers various ossibilities for interesting inquiry questions for students, terefore encouraging discussion and a searc for te rigt answers. For tese reasons we decided to build a large-scale version of te Cartesian diver for our ands-on science centre in Ljubljana. But, as in any oter science centre were eole build teir own exibits, we also wanted to add someting new to our exibit. Te following idea emerged from discussion: let s try wit two liquids of different densities tat do not mix and two or more divers tat will initially float at different liquid boundaries. As usual, te idea was first tested on a simle rototye made from material tat one can find easily. Wile laying wit a rototye it aeared to one of us tat te two-liquid Cartesian diver may be useful as a demonstration exeriment in scool. In tis article we resent te construction of te exeriment and some suggestions for ow to use it at te advanced level in secondary scool. Tere are numerous references in te literature and on te web on ow to construct a simle Cartesian diver, ow to exlain its beaviour and ow to use it in building students ideas about te observed enomena. Wen searcing for revious reorts on multi-liquid Cartesian divers we found only a reort on te Cartesian diver using continuously varying liquid density obtained wit salted water [1]. One drawback of suc a diver is tat even if we do not touc te bottle te density of te liquid will eventfully become uniform due to convection and diffusion. How to make it Fill alf of a soda-o bottle wit a liquid for wasing car windows (we used Sonax, te etanolbased, blue-coloured liquid wit a density of about 0.91 g cm 3 tat freezes at 40 C according to te roducer) and te rest wit araffin oil, a colourless, odourless liquid wit a density of about 0.86 g cm 3. Fill te araffin oil almost to te to of te bottle. Te different colours, similar viscosity and te fact tat tey do not mix togeter make tese two liquids ideal for our exeriment. In addition, te two liquids do not deteriorate wit time. However, te simle combination of water and cooking oil also works well, as sown later in tis article. Making te divers Tere are several suggestions in te literature and on te web for ow to make divers using eyedroers, test tubes or even ketcu bags. We made ours from folded drinking straws and we 58 P HYSICS E DUCATION 39 (1) /04/ $ IOP Publising Ltd

2 Two-liquid Cartesian diver (a) (b) (c) (d) Figure 1. Two-stage Cartesian diver: (a) wen left alone; (b) (d) as te lastic bottle is squeezed arder and arder. used crocodile clis to adjust teir masses (see figure 1). Adjusting te divers First take a iece of tick wire tat is about 5 cm longer tan te eigt of te bottle, and bend it to make a ook at one end. You will need it for lifting te sinking divers from te bottle. Make a first diver as exlained in te revious aragra and ut it into te filled bottle. If you are lucky te diver sould float on te uer liquid or on te lower liquid, at te boundary between te two liquids. Deending on tis, cut a second diver s straw so tat it will float on te surface of te oter liquid (if you cut a longer straw te diver will be less dense and vice versa). Be reared to make fine adjustments by cutting off sort ieces of te straws to finally acieve te desired result (figure 1(a)). Any diver tat sinks to te bottom of te bottle sould be relaced by a new one made from a longer iece of straw. Wen you ave adjusted te divers, close te bottle and start squeezing it wit your fingers. In our case, te lower diver sank first (figure 1(b)). Wen te bottle was squeezed furter te uer diver started sinking but stoed at te boundary between te liquids (figure 1(c)). By varying te force on te bottle walls te diver sank more or less into te lower liquid, as its effective density was canged. By squeezing te bottle even arder te uer diver eventually also sank to te bottom (figure 1(d)). Reducing te ressure on te bottle walls caused te divers to rise to teir initial January 2004 P HYSICS E DUCATION 59

3 G Planinšič et al Figure 2. Te kitcen version of te two-liquid Cartesian diver: cooking oil at te to, water at te bottom and ketcu bags as divers. See text for details on ow to adjust te divers. ositions in te reverse order to tat in wic tey sank. Kitcen version of te two-liquid Cartesian diver Te two-liquid Cartesian diver can also be made from materials found in te kitcen. Cooking oil and water make good substitutes for te araffin oil and Sonax. As exlained earlier, ketcu bags are ideal divers. Te little air bubble traed in te bag works in te same way as te air bubble traed in te straw. However, fine adjustment of te ketcu divers takes a little more atience. Te lower one can be made eavier by fixing one or two aer clis to it. Te uer one sould be made a little ligter. Tis can be acieved by gluing a narrow stri of Styrofoam to it. Te kitcen version of te two-liquid Cartesian diver in oeration is sown in figure 2. It is instructive to exlore and comare te divers beaviour wen tey are bot immersed in te uer or te lower liquid wit tat wen tey are between te two liquids. Using te two-liquid Cartesian diver in te ysics classroom Here are some suggestions for inquiry questions tat can be asked wen sowing te two-stage Cartesian diver in scool: Q1. Plot a gra tat will sow ow te density of te liquid in te bottle varies wit te det. How does te gra cange wen te bottle is squeezed? Q2. Plot a gra tat will sow ow te ressure in te bottle varies wit det. (Note tat tere is a small air bubble traed in te bottle.) How does te gra cange wen te bottle is squeezed? Q3. Try to redict ow a cange of temerature would affect te exeriment (recall ow te Galileo termometer works). Te answers to te first two questions are sown in figure 3 but readers are encouraged to find and exerimentally ceck te answer to te last question. We ave demonstrated te two-liquid Cartesian diver and osed questions Q1 and Q2 to a grou of 31 first-year ysics students (age 19) at one of te first meetings at te beginning of te scool year (in Slovenia ysics is a comulsory subject at all secondary scools). Students were reminded tat te liquids are ractically incomressible. About 26% of te answers were comletely correct and 45% of te students gave te correct answer to Q1. Tere were no cases wit bot te wrong answer to Q1 and te correct answer to Q2. Some tyical wrong answers are 60 P HYSICS E DUCATION January 2004

4 Two-liquid Cartesian diver 3. Density and ressure Figure 3. Te liquid density and te ressure in te bottle as a function of det in te closed bottle filled wit two liquids tat do not mix. Te dased line sows te cange after te bottle is squeezed. f 1 f 2 sown in figure 4. It was clear tat many students ad roblems transforming te natural vertical axis defined by te direction of g in te exeriment into te orizontal axis on te gra. We did not analyse te results in detail but it was interesting tat te ercentage of comletely correct answers (26%) matces very well wit te tyical ercentage of first-year students wo manage to comlete te first-year exams in our deartment in te first term. A systematic analysis of student understanding of Arcimedes rincile as recently been ublised elsewere [2]. Using te two-liquid Cartesian diver in building te model At te beginning of a well-known textbook one can read, Models are simle artificial worlds created to give insigt into ow real systems work, and redict wat tey migt do. We start wit te simle models... and go on to model variations of one quantity wit anoter [3]. As exlained later in te same book, te evolution of a model goes troug several stes before te model is acceted. Te derivation of a teoretical model for te Cartesian diver, wic floats between two liquids, may be a good examle for secondary scool students at advanced level. Observations Everyting starts wit te observations. Let s concentrate on te diver tat initially floats at te boundary between te two liquids. Part of it is in te uer fluid and te rest is submerged in te lower fluid. If I squeeze te bottle gently (and kee te force on te bottle constant), te diver moves down a little and finds a new equilibrium osition. If I continue squeezing te bottle te wole diver eventually sinks into te lower liquid and down to te bottom of te bottle. 4. Wrong answers /2 (a) (b) (c) /2 Figure 4. Tyical wrong answers to te questions Q1 and Q2 (see te text for details). Te dased line sows te redicted cange after te bottle was squeezed. In (c) no cange after squeezing as been redicted. January 2004 P HYSICS E DUCATION 61

5 G Planinšič et al Building a teoretical model We wis to ave a teoretical model (a formula, if you wis) tat will correctly exlain te beaviour of te diver between te two liquids. Te imortant art of building te model is to decide wat arameters or enomena make a major contribution to our exeriment and wat can be neglected. Te usual aroac is to make te first model as simle as ossible and see if it suorts te observations. If it doesn t, try to take into account wat was neglected in te first model (one arameter or enomenon at a time, te easiest first) and ceck te model again. Using te aroriate questions, students may be guided to come to te following assumtions tat will lead tem to build te simlest model for te diver between te two liquids (for clarity we list te assumtions first but in ractice it is better to bring tem u during te derivation of te model): Te liquids are ractically incomressible, so teir densities are constant and do not cange wit det. Te diver consists of te straw, te crocodile cli and te air bubble traed in te straw. It is reasonable to say tat te volume of te air bubble is bigger tan te sum of te volumes of te crocodile cli and te straw. In our first crude model we will terefore assume tat te crocodile cli and te straw (i.e. te lastic) ave negligible volumes and tat te volume of te diver V is aroximately equal to te volume of te air bubble. It is imortant to emasize ere tat te mass of te diver m is also equal to te sum of te tree masses but now te mass of te crocodile cli is te largest. However, as te total mass of te diver does not cange during te exeriment, no aroximation needs to be made ere. Te eigt of te diver is less tan 10 cm. Te corresonding cange in ydrostatic ressure during te diver s excursion is terefore less tan 1/50 of te normal ambient ressure. In te simle model we will assume tat te contribution of te ydrostatic ressure to te ressure felt by te traed air bubble may be neglected 1. In oter words, we assume tat te air bubble feels only te ambient ressure 1 Tis assumtion can be suorted by te estimation of te added ressure roduced by our fingers needed to sink te diver (measure te force and estimate te contact area). lus te ressure exerted on te walls of te soda bottle by our fingers. However, note tat te small difference in ydrostatic ressure is essential in exlaining te buoyancy force on te diver! Te temerature of te liquids and te air traed in te straw is constant during te exeriment. Let s see wat forces act on te diver tat floats between te two liquids. Since te diver is at rest, te sum of te forces sould be zero. Te total buoyancy force can be seen as te sum of two contributions. Te first is from te art of te diver wit volume V 1 tat is submerged in te uer liquid wit density 1 ; te second is from te rest of te diver of volume V 2 tat is submerged in te lower liquid wit density 2 (obviously 2 > 1 ). Te total buoyancy force is balanced by te weigt of te diver. Te same statements can be formulated matematically as follows: mg + F buoy = 0 (1) were F buoy = 1 gv gv 2 (2) V = V 1 + V 2. (3) Te last equality is one way of saying, Te volume of te diver consists of te two arts V 1 and V 2. Anoter way of describing te same ting is by writing te following two equations: V 1 = ηv V 2 = (1 η)v (4) were η denotes te fraction of te total volume of te diver tat is in te uer liquid. For examle, if we say, one quarter of te diver is in te uer liquid and te rest (i.e. tree quarters) is in te lower liquid, ten η = Obviously 0 η 1. Note tat our main goal is to obtain an exression tat will describe ow η deends on te ressure in te bottle. Using te equations above one can exress te dive ratio η as η = 2 m/v 2 1. (5) Since we assumed tat te temerature remains constant troug te exeriment and tat te diver s volume is aroximately equal to te volume of te air bubble, we can use Boyle s law to 62 P HYSICS E DUCATION January 2004

6 Two-liquid Cartesian diver Figure 5. Two divers of different sizes cross te boundary between te two liquids as te ressure in te bottle increases (from left to rigt). Te divers are made from 5 mm diameter drinking straws, glued at te to and weigted at te oen end. Initially te divers were barely floating on te to of te uer liquid. M An MPEG movie of tis figure is available from stacks.io.org/ysed/39/58 relate te volume of te air bubble to te ressure in te bottle. In our case Boyle s law reads as 0 V 0 = ( 0 + )V (6) were 0 is te normal ambient ressure (about 10 5 Nm 2 ), V 0 is te initial volume of te air bubble and is te additional ressure caused by squeezing te bottle wit our fingers. Now equation (5) can be written in te final form η = 2 eff 0 (1+/ 0 ) (7) 2 1 were eff 0 is equal to te ratio m/v 0 and terefore lays te role of te effective density of te diver at te beginning of te exeriment. Te diver was initially adjusted to sink in te uer liquid and float between te two liquids, so we know tat 2 > eff 0 > 1. Equation (7) is suosed to describe in a matematical way ow te osition of te diver canges wit te ressure in te bottle. In order to trust te equation and justify te model we ave to verify weter te equation redicts correctly wat as been observed. In scool tis verification is usually done on a few simle cases as sown in two stes below. 1. Observation: If te bottle is not touced, te diver floats between te two liquids. Teoretical rediction based on equation (7): = 0givesη = ( 2 eff 0 )/( 2 1 ). Taking into account te relationsi between te tree densities, one finds tat indeed η < 1, wic agrees wit te observation. 2. Observation: Wen I squeeze te bottle arder, te diver moves down into te lower liquid. At a certain ressure te wole diver sinks into te lower liquid. Teoretical rediction based on equation (7): Wen increases, te numerator in te equation decreases (te denominator is constant) and terefore η decreases, wic agrees wit te observation. At a certain ressure η = 0. Tis value can be calculated from equation (7) and is equal to ( ) 2 = 0 1. (8) eff 0 Tis is te ressure needed to sink te diver comletely into te lower liquid, as redicted by te model. If aroriate equiment is available, te calculated value can be comared wit te measured value. It is imortant to note tat once te wole diver is in te uer or lower liquid, η becomes constant January 2004 P HYSICS E DUCATION 63

7 G Planinšič et al (1 or 0 resectively) and is no longer given by equation (7). Tat is not quite te end! Sometimes studying and verifying te model can lead to te rediction of a new exerimental result tat we ad not tougt about before. Of course, we sould know tat since te model is only an idealization of te real situation suc a rediction migt be wrong. But in our case it aened to work well. Note tat in equation (7) all te information about te diver is concentrated in eff 0. Terefore tis equation describes te sinking of all te divers on tis world tat ave te same initial effective density! Or in oter words, all divers, no matter ow big or small, will cross from one liquid to te oter togeter, roviding tat tey ave te same initial density. Tis rediction can be verified exerimentally by using two divers of different sizes and adjusting teir masses so tat te divers initially barely float on te uer liquid. Te result of te exeriment is sown in figure 5 but can be also watced as a movie on te journal s website (see stacks.io.org/ysed/39/58). Received 28 October 2003 PII: S (04) DOI: / /39/1/003 References [1] Cosby R M and Petry D E 1989 Simle buoyancy demonstrations using saltwater Pys. Teacer [2] Loverude M E, Kautz C H and Heron PRL2003 Am. J. Pys [3] Ogborn J and Witeouse M (ed) 2001 Advancing Pysics A2 (Bristol: Institute of Pysics Publising) Gorazd Planinšič received is PD in ysics from te University of Ljubljana, Slovenia. Since 2000 e as led te undergraduate Pedagogical Pysics course and ostgraduate course on Educational Pysics at te University. He is co-founder and collaborator of te Slovenian ands-on science centre Te House of Exeriments and as also been secretary of GIREP since Mia Kos received is PD in ysics from te University of Ljubljana in Since 1996 e as been te director and co-founder of first Slovenian ands-on science centre called Hisa ekserimentov (House of Exeriments). He is also te cief editor of a cildren s magazine for early education called Petka. Riko Jerman teaces ysics at te tecnical scool in Ljubljana. He graduated in 1977 from te University of Ljubljana were e also got a Master s degree in cemistry. He is also involved in develoing new science textbooks for elementary level and in designing exeriments for te Ljubljana science centre House of Exeriments. 64 P HYSICS E DUCATION January 2004