Floating between two liquids. Laurence Viennot.

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1 Floating between two liquids Laurence Viennot Abstract The hydrostatic equilibrium of a solid floating between two liquids is analysed, first as a classical exercise, then as a simple experiment with a particular staging. It is suggested to use this situation to address some students difficulties about: pressure in statics of fluids, role of atmospheric pressure, Archimedes up-thrust, and thinking this setting as a system. It is shown that a possible way to help students overcome some of these difficulties, especially the last one, is to use graphical representations. A very classical exercise is the analysis of the hydrostatic equilibrium of a cylindrical body height H, area of the base S, mean density ρ s - situated in a recipient with two immiscible liquids of density, respectively ρ 1 and ρ 2, such that ρ 1 > ρ s > ρ 2. The body floats in the first one alone (ρ 1 > ρ s ) and sinks in the other one alone (ρ s > ρ 2 ). The hydrostatic equilibrium of the solid between two liquids may be like one of the two cases (a or b) represented in Figure 1. Liquid 2 h 2 h 1 H Liquid 2 h 2 h 1 H Liquid 1 Liquid 1 a The floating body is covered by liquid 2. b The floating body is not covered by liquid 2 Then, the body is in contact with the air. Figure 1 Two cases for a cylindrical solid at hydrostatic equilibrium between two liquids. 1

2 Materials The materials needed to implement this experiment are very simple, for example. - A transparent cylindrical glass filled with water (possibly with a colorant). - Groundnut oil, usually about 50 cm 3 are sufficient.. -A plastic egg (toys, e.g. Kinder) or any empty tube of pills, to be partly filled with dense materials (e.g. coins, shot) until the floating in water (alone) is ensured. It is in a stable vertical position, the top of the solid emerging by about the third or the quarter of its height H out the free surface of water. Classical solution For any position of the solid, the relationship of fluid statics p = - ρ g z can be used for each part of the cylinder immersed in each liquid, of respective heights h 1 and h 2. Therefore, with an upward axis, the differences in pressure between lower and upper horizontal sections of the cylinder immersed in, respectively, liquid 1 and 2 are given by: p 1 = ρ 1 gh 1 p 2 = ρ 2 gh 2 The possible contribution of the air (case b in Fig. 1) can be neglected with respect to the two others, given that the density of the air is typically a thousand times smaller than those of the liquids. A state of equilibrium occurs when Archimedes up-thrusts due to each liquid and the weight of the body balance out: (ρ 1 h 1 + ρ 2 h 2 ) Sg - ρ s HS g = 0 or else: ρ 1 h 1 + ρ 2 h 2 = ρ s H Solving for, say, h 1 gives h 1 = (ρ s H - ρ 2 h 2 )/ρ 1 (1) h 1 and h 2 being the heights of each part of the cylinder immersed in each liquid in case the cylinder actually floats between the liquids. In particular, relationship (1) gives the value of h 1 that is necessary for the body to float in liquid 1 alone (then h 2 = 0), or to be in a state of equilibrium of the type floating between two liquids with a given value of h 2. Would the actual volume of liquid 1 be too small to ensure this condition on h 1, then the body would rest on the bottom of the recipient. The statements of this section, and in particular relationship (1), hold for case a and case b in Figure 1; with, in case b, the approximation that Archimedes up-thrust due to air is neglected with respects to the two other contributions. In case a, when the body is covered by fluid 2, h 1 + h 2 = H and relationship (1) leads to h 1 = h 2 (ρ s -ρ 2 ) / (ρ 1 -ρ s ) (2) or else h 1 = H (ρ s -ρ 2 ) / (ρ 1 -ρ 2 ) (3) Stability of the equilibrium between two fluids : For the cylindrical solid to stay in a stable vertical position, it is necessary that the center of mass of the solid be lower than the center of 2

5 Staging the situation in order to stress the systemic aspect Among the facilitating tools that can be used with students, beyond the classical solution, the graphical approach is of interest A graph can be constructed, showing pressure against altitude without (black line in fig. 2) and with (coloured line in fig. 2) the second liquid. The approximation done previously about the role of the air - i.e. a negligible contribution to Archimedes up-thrust - has a graphical equivalent: a constant value of atmospheric pressure near the recipient. What counts to evaluate the up-thrust on the cylinder is the difference between the pressure forces exerted on the lower and upper horizontal surfaces of this body, situated at a given altitude interval H. For any hydrostatic equilibrium, this difference in pressure, p eq, is such that S p eq =g (ρ s S H) or else p eq = g ρ s H p In black: only one liquid In colour: two liquids Surface of liquid 1 Surface of liquid 2 p p eq p atm Case a: H = h 1 +h 2 h 1 h 2 z Case b: H > h 1 +h 2 Fig. 2 Pressure versus altitude in two situations: only liquid1 (black), liquid 2 added on top of liquid 1 (coloured). When the second liquid is added the solid will move upwards (due to increased p), reaching a new equilibrium: see Fig. 3. It can be seen that, at the initial equilibrium position, the difference in pressure p between the lower and the upper horizontal surface limiting the cylinder is larger when the second liquid is added. Hence the resulting increased up-thrust. The body then moves upwards and reaches a new equilibrium position, where p retrieves its initial value (fig. 3). 5

6 An important point to be stressed is that the effect of adding the second fluid on top of the first one is a change of the whole field of pressure: a systemic view. As regards Figure 2, some possibly useful questions to ask the students may be: - explain, in your words, why, concerning a part of the liquid, the coloured and black lines are parallel; - explain the physical meaning of their distance when these lines are parallel; - explain what happens when the atmospheric pressure increases; - explain what happens if the two liquids have the same density (the case when liquid 1 = liquid 2. All the various cases (not enough water to ensure floating only in liquid 1, case a and b in Figure 1, density ρ 2 lower than ρ 1 and larger than ρ s ) can be discussed in terms of the difference in pressure p between lower an upper sections of the solid. Performing all the corresponding experiments after justified predictions will feed all the discussions about these various cases. p p eq H Surface of liquid 1 Surface of liquid 2 p eq p atm Initial position of the base of the cylinder h 1 h 2 H z Fig. 3 Looking for the new equilibrium. For given values of H and p eq -, the set square in the top right angle should be moved and stuck against the coloured line representing the field of pressure. This gives at the same time the new position of the body and the new values of the heights of the immersed parts. Practical suggestion: the set square Finding any equilibrium position sums up in fitting: -the field of pressure, graphically represented, and characteristic of the fluid. -two characteristics of the body: height H and p eq ( p eq = g ρ s H) 6

7 This can be highlighted by using a cardboard set-square and fitting it on a graph drawn on the blackboard, like in fig. 3. Many different situations can be solved with this technique. Using the set square also makes it possible to predict qualitatively the force exerted by the fluids on the cylinder when this body is pushed downward by a distance d from its equilibrium position. Note that the order of magnitude of atmospheric pressure, compared with that of pressure differences in this situation (x100), does not allow to set the origin of pressure axis at the intersection of the axes. Links with other situations Stressing that adding the second fluid on top of the first one changes the whole field of pressure comes down to emphasize the need for a systemic analysis. This essential idea is also very useful to overcome common ideas such that: pressure is directly linked to the weight of the column of fluid that is above your head. For instance, it is easy to evidence the corresponding risks of misunderstanding with the two-fish situation (fig. 3, see Pugliese-Jona 1984), previously discussed in the paper Various experiments involving fluid statics (Viennot et al., MUSE). As documented by Besson (Besson 2004, Besson & Viennot 2004, Viennot 2003), a large proportion of students tend to think that pressure is lower in an underwater cave (fish 1) than in the open sea at same depth (fish 2), as if the height of water above the cave s ceiling affected pressure only in the open sea. Figure 3 Two-fish situation The two-fish problem is also an opportunity to stress the fact that Archimedes up-thrust should be seen, first of all, as the outcome of differences in values of pressure in the fluid. All too often, the mere use of the formula screens this central aspect. Still more generally, the importance of differences in physics is so crucial that many other situations deserve the same spotlighting (see for instance Staging the siphon (Viennot &Planinsic 2009) on this MUSE site. 7

8 An activity with a slightly more complex situation a Cartesian diver between two fluids has been proposed and analysed by G. Planinsic, M. Kos and R. Jerman (2004). Then, the transformation envisaged is to squeeze the container and discuss how the state of equilibrium is changed. The main focus is then on the change of volume of the diver, but an analysis of the changes in pressure is also suggested, as well as the students difficulties in this respect. References Bennhold, C. & Feldmann, G Instructor Notes On Conceptual Test Questions, In Giancoli Physics- Principle with applications, 6 th Edition, Pearson, Prentice Hall, Besson, U. & Viennot, L Using models at mesoscopic scale in teaching physics: two experimental interventions on solid friction and fluid statics, International Journal of Science Education, 26 (9), Besson, U Students conceptions of fluids. International Journal of Science Education, 26, Closset J.-L Les obstacles à l'apprentissage de l'électrocinétique. Bulletin de l'union des Physiciens, 716, Fauconnet, S Etude de résolution de problèmes. Quelques problèmes de même structure. Thèse de troisième cycle, Université Paris 7. Planinsic, G., Kos, M. & Jerman, R Two-liquid Cartesian diver. Physics Education, 39(1), Pugliese-Jona, S Fisica e Laboratorio, vol. 1, Turin: Loescher. Rozier, S. & Viennot, L Students' reasoning in elementary thermodynamics, International Journal of Science Education, 13 (2), Shipstone, D Electricity in simple circuits. Children's Ideas in Science. Open University Press, Milton Keynes, Viennot L Reasoning in Physics The part of common sense, Dordrecht: Kluwer Ac.Pub. Viennot L Teaching physics. Dordrecht: Kluwer Ac. Pub. Viennot, L. & Planinsic, G Staging the siphon, MUSE Viennot, L., Planinsic, G., Sassi, E. & Ucke, C Various experiments involving fluid statics, MUSE 8

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