Mathematical Analysis of Archimedes' Principle: Validity and Exceptions, Slides of Law

Download Mathematical Analysis of Archimedes' Principle: Validity and Exceptions and more Slides Law in PDF only on Docsity! ar X iv :1 11 0. 52 64 v1 [ ph ys ic s. cl as s- ph ] 2 4 O ct 2 01 1 Using surface integrals for checking the Archimedes’ law of buoyancy F M S Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia-DF, Brazil E-mail: fabio@fis.unb.br Abstract. A mathematical derivation of the force exerted by an inhomogeneous (i.e., compressible) fluid on the surface of an arbitrarily-shaped body immersed in it is not found in literature, which may be attributed to our trust on Archimedes’ law of buoyancy. However, this law, also known as Archimedes’ principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a downward force that increases with depth is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a continuous function of depth. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact. PACS numbers: 01.30.mp, 01.55.+b, 47.85.Dh Submitted to: Eur. J. Phys. Accepted for publication: 10/20/2011 Checking the validity of Archimedes’ law 2 1. Introduction The quantitative study of hydrostatic phenomena did begin in antiquity with Archimedes’ treatise On Floating Bodies - Book I, where some propositions for the problem of the force exerted by a liquid on a body fully or partially submerged in it are proved [1, 2].‡ In modern texts, his propositions are reduced to a single statement known as Archimedes’ law of buoyancy, or simply Archimedes’ principle (AP), which states that “any object immersed in a fluid will experience an upward force equal to the weight of the fluid displaced by the body” [3, 4]. The continuity of this work had to wait about eighteen centuries, until the arising of the scientific method, which was the guide for the experimental investigations in hydrostatics by Stevinus, Galileo, Torricelli, and Pascal, among others.§ The very long time interval from Archimedes to these experimentalists is a clear indicative of the advance of Archimedes thoughts. As pointed out by Netz (based upon a palimpsest discovered recently), Archimedes developed rigorous mathematical proofs for most his ideas [5]. However, the derivation of the exact force exerted by an inhomogeneous fluid on an arbitrarily-shaped body immersed in it, as will be shown here, demands the knowledge of the divergence theorem, a mathematical tool that was out of reach for the ancients. Therefore, the validity of the Archimedes propositions for this more general case was not formally proved on his original work. By defining the buoyant force (BF) as the net force exerted by a fluid on the portion of the surface of a body (fully or partially submerged) that touches the fluid, the validity of AP in predicting this force can be checked. In fact, the simple case of symmetric solid bodies (e.g., a right-circular cylinder, as found in Ref. [6], or a rectangular block, as found in Ref. [7]) immersed in a liquid is used in most textbooks for proofing the validity of AP. There, symmetry arguments are taken into account to show that the horizontal forces exerted by the liquid cancel and then the net force reduces to the difference of pressure forces exerted on the top and bottom surfaces [8]. The BF is then shown to point upwards, with a magnitude that agrees to AP, which also explains the origin of the BF in terms of an increase of pressure with depth [9]. Note, however, that this proof works only for symmetric bodies with horizontal, flat surfaces on the top and the bottom, immersed in an incompressible (i.e., homogeneous) fluid. Although an extension of this result for arbitrarily-shaped bodies immersed in a liquid can be found in some textbooks [10, 11, 12], a formal generalization for bodies immersed in a compressible (i.e., inhomogeneous) fluid is not found in literature. This certainly induces the readers to believe that it should be very complex mathematically, which is not true, as will be shown here. ‡ Certainly in connection to the need, at that time of flourishing shipping by sea routes, of predicting how much additional weight a ship could support without sinking. § Gravesande also deserves citation, due to his accurate experiment for comparing the force exerted by a liquid on a body immersed in it to the weight of the displaced liquid. This experiment uses a bucket and a metallic cylinder that fits snugly inside the bucket. By suspending the bucket and the cylinder from a balance and bringing it into equilibrium, one immerses the cylinder in a water container. The balance equilibrium is then restored by filling the bucket with water. Checking the validity of Archimedes’ law 5 does not appear to be tractable analytically due to the dependence of the direction of n̂ on the position r over S, which in turn depends on the (arbitrary) shape of S. However, this task can be easily worked out for a body fully submerged in a homogeneous fluid (ρ = const.) by applying the divergence theorem to the vector field E = −p(z) k̂, which yields a volume integral that evaluates to ρ g V k̂, in agreement to AP, as discussed in some advanced texts [10, 24]. For the more general case of a body fully or partially submerged in an inhomogeneous fluid (or a set of fluids), I shall follow a slightly different way here, based upon the following version of the divergence theorem [10, 23, 25]. Gradient theorem. Let R be a bounded region in space whose boundary S is a closed, piecewise smooth surface which is positively oriented by a unit normal vector n̂ directed outward from R. If f = f(r) is a scalar function with continuous partial derivatives in all points of an open region that contains R (including S), then ∫∫ S © f(r) n̂ dS = ∫∫∫ R ∇f dV . In Appendix A, it is shown how the divergence theorem can be used for proofing the gradient theorem. The advantage of using this less-known calculus theorem is that it allows us to transform the surface integral in Eq. (7) into a volume integral of ∇p , a vector that can be easily written in terms of the fluid density via the hydrostatic equation, Eq. (4). As we are only interested in pressure forces, let us substitute f(r) = −p(r) in both integrals of the gradient theorem. This yields − ∫∫ S © p(r) n̂ dS = − ∫∫∫ Vf ∇p dV , (8) where the surface integral at the left-hand side is, according to our definition, the BF itself whenever the surface S is closed, i.e. when the body is fully submerged in a fluid. Let us analyze this more closely. 2.1. A body fully submerged in a fluid For a body of arbitrary shape fully submerged in a fluid (or a set of fluids), by substituting the pressure gradient in Eq. (4) on Eq. (8), one finds [36] F = − ∫∫∫ Vf ∇p dV = [∫∫∫ Vf ρ(z) dV ] g k̂ . (9) For the general case of an inhomogeneous, compressible fluid whose density changes with depth, as occurs with gases and high columns of liquids [26], the pressure gradient in Eq. (9) will be integrable over V as long as ρ(z) is a continuous function of depth (in conformity to the hypothesis of the gradient theorem). Within this condition, one has F = [∫∫∫ Vf ρ(z) dV ] g k̂ = [∫∫∫ V ρ(z) dV ] g k̂ . (10) Checking the validity of Archimedes’ law 6 Since ∫∫∫ V ρ(z) dV is the mass mf of fluid that would occupy the volume V of the body (fully submerged), then F = mf g k̂ , (11) which is an upward force whose magnitude equals the weight of the fluid displaced by the body, in agreement to AP as stated in Eq. (1). This shows that AP remains valid even for an inhomogeneous fluid, as long as the density is a continuous function of depth, a condition fulfilled in most practical situations. 2.2. A body partially submerged in a fluid The case of an arbitrarily-shaped body floating in a fluid with a density ρ1(z), with its emerged part exposed to either vacuum (i.e., a fictitious fluid with null density) or a less dense fluid is an interesting example of floating in which the exact BF can be compared to the force predicted by AP.¶ This is important for the study of many floating phenomena, from ships in seawater to the isostatic equilibrium of tectonic plates (known in geology as isostasy) [37]. Without loss of generality, let us restrict our analysis to two fluids, one (denser) with a density ρ1(z), we call fluid 1, and another (less dense) with a density ρ2(z) ≤ min [ρ1(z)] = ρ1(0 −), we call fluid 2. For simplicity, I choose the origin z = 0 at the planar surface of separation between the fluids, as indicated in Fig. 2, where the fluid density can present a discontinuity ρ1(0 −)− ρ2(0 +). The forces that these fluids exert on the body surface can be evaluated by applying the gradient theorem to each fluid separately, as follows. First, divide the body surface S into two parts: the open surface S1 below the interface at z = 0 and the open surface S2, above z = 0. The integral over the (closed) surface S in Eq. (7) can then be written as ∫∫ S © p(z) n̂ dS = ∫∫ S1 p(z) n̂1 dS + ∫∫ S2 p(z) n̂2 dS , (12) where n̂1 (n̂2) is the outward unit normal vector at a point of S1 (S2), as indicated in Fig. 2. Let us call S0 the planar surface, also indicated in Fig. 2, corresponding to the horizontal cross-section of the body at z = 0. By noting that n̂1 = k̂ and n̂2 = −k̂ in all points of S0, then, being p(z) a continuous function, one has ∫∫ S0 p(z) n̂1 dS + ∫∫ S0 p(z) n̂2 dS = 0 . This allows us to use S0 to generate two closed surfaces, S̃1 and S̃2, formed by the unions S1 ∪ S0 and S2 ∪ S0, respectively. From Eq. (12), one has ∫∫ S © p(z) n̂ dS = ∫∫ S1 p(z) n̂1 dS + ∫∫ S0 p(z) n̂1 dS + ∫∫ S2 p(z) n̂2 dS + ∫∫ S0 p(z) n̂2 dS ¶ A null pressure is assumed on the portions of S that are not interacting with any fluid. Checking the validity of Archimedes’ law 7 = ∫∫ S̃1 © p(z) n̂1 dS + ∫∫ S̃2 © p(z) n̂2 dS . (13) As both S̃1 and S̃2 are closed surfaces, one can apply the gradient theorem to each of them, separately. This gives F = −   ∫∫ S̃1 © p(z) n̂1 dS + ∫∫ S̃2 © p(z) n̂2 dS   = − (∫∫∫ V1 ∇p dV + ∫∫∫ V2 ∇p dV ) = − (∫∫∫ V1 ∂p ∂z dV + ∫∫∫ V2 ∂p ∂z dV ) k̂ , (14) where V1 and V2 are the volumes of the portions of the body below and above the interface at z = 0, respectively. From the hydrostatic equation, one has ∂p/∂z = −g ρ(z), which reduces the above integrals to∫∫∫ V1 [−ρ1(z) g] dV + ∫∫∫ V2 [−ρ2(z) g] dV = −g [∫∫∫ V1 ρ1(z) dV + ∫∫∫ V2 ρ2(z) dV ] . (15) The latter volume integrals are equivalent to the masses m1 and m2 of the fluids 1 and 2 displaced by the body, respectively, which reduces the BF to F = g [∫∫∫ V1 ρ1(z) dV + ∫∫∫ V2 ρ2(z) dV ] k̂ = (m1 +m2) g k̂ . (16) The BF is then upward and its magnitude is equal to the sum of the weights of the fluids displaced by the body, in agreement to AP in the form given in Eq. (1). Note that the potential energy minimization technique described in Refs. [27, 28, 29] cannot provide this confirmation of AP because it works only for rigorously homogeneous (i.e., incompressible) fluids. Interestingly, our proof shows that the exact BF can also be found by assuming that the body is fully submerged in a single fluid with a variable density ρ(z) that is not continuous, but a piecewise continuous function with a (finite) leap discontinuity at z = 0.+ Although the contribution of fluid 2 to the BF is usually smaller than that of fluid 1, it cannot in general be neglected, as done in introductory physics textbooks [30]. In our approach, this corresponds to assume a constant pressure on all points of the surface S2 of the emerged portion, which is incorrect. This leads to a null gradient of pressure on the emerged part of the body, which erroneously reduces the BF to only F =   ∫∫∫ V1 ρ1(z) dV   g k̂ . (17) + Interestingly, this suggests that a more general version of the divergence theorem could be found, in which the requirement of continuity of the partial derivatives of f(r) (respectively, of ∇ · E) could be weakened to only a piecewise continuity. I have not found such generalization in literature. Checking the validity of Archimedes’ law 10 account by those interested in to develop a downward BF experiment similar to that proposed by Bierman and Kincanon [15], since a part of the bottom of the block with an area Aair is intentionally left in contact to air (this comes from their technique to reduce the liquid seepage under the block) [17]. This changes the BF to F = − (pb A− p0Aair − ρ V g) k̂ . (23) For an arbitrarily-shaped body immersed in a liquid (in the bottom case), let us assume that there is a non-null area Ab of direct contact between the body and the bottom of a container. If no liquid seeps under the block, then the pressure exerted by the liquid there at the bottom of the body is of course null. The BF is then F = − ∫∫ S2∪Ab p(z) n̂ dS = − [∫∫ S2 p(z) n̂ dS + ∫∫ Ab 0 ( −k̂ ) dS ] = − ∫∫ S2 p(z) n̂ dS . In view to apply the gradient theorem, one needs a surface integral over a closed surface. By creating a fictitious closed surface Σ = S2 ∪ Ab on which the pressure forces will be exerted as if the body would be fully surrounded by the liquid (i.e., one assumes a constant pressure pb over the horizontal surface Ab), one has F = − ∫∫ S2 p(z) n̂ dS − ∫∫ Ab pb ( −k̂ ) dS + ∫∫ Ab pb ( −k̂ ) dS = −   ∫∫ Σ © p(z) n̂ dS − ∫∫ Ab pb ( −k̂ ) dS   = − ∫∫ Σ © p(z) n̂ dS − pb k̂ ∫∫ Ab dS = ρ V g k̂− pb Ab k̂ = − (pb Ab − ρ V g) k̂ . (24) This is again a downward BF that increases linearly with depth, since pb = p0 + ρ gH , H being the height of the liquid column above the bottom, as indicated in Fig. 3. This result for arbitrarily-shaped bodies is not found in literature. Incidentally, this result suggest that the only exceptions to AP, for fluids in equilibrium, are those cases in which ∇p is not a piecewise continuous function over the whole surface Σ of the body, otherwise the results of the previous section guarantee that the BF points upward and has a magnitude ρ V g, in agreement to AP. This includes all contact cases, since the boundary of the contact region is composed by points around which the pressure leaps (i.e., changes discontinuously) from a strictly positive value p(z) (at the liquid side) to an smaller (ideally null) pressure (at the contact surface). Therefore, in these points the pressure is not a differentiable function (because it is not even a continuous function), which impedes us of applying the gradient theorem [38]. 4. Conclusions Here in this paper, I have drawn the attention of the readers to the fact that the BF predicted by AP can be derived mathematically even for bodies of arbitrary shape, fully Checking the validity of Archimedes’ law 11 or partially submerged in a fluid, homogeneous or not, based only upon the validity of the hydrostatic equation and the gradient theorem. For that, I first define the buoyant force as the net force exerted by a fluid on the portion of the surface of the body that is pressed by the fluid. Then, the exact BF becomes a surface integral of the pressure force exerted by the fluid, which can be easily evaluated when the body is fully surrounded by the fluid. In this case, the gradient theorem allows one to convert that surface integral into a volume integral which promptly reduces to an upward force with a magnitude equal to the weight of the displaced fluid (as predicted by AP), as long as the fluid density is a continuous function of depth. Finally, some cases were pointed out in which AP fails and this could help students (even teachers) to avoid erroneous applications of this physical law. The exact force in one of these exceptional cases is determined here by applying our surface integral approach to a body immersed in a liquid and in contact to the bottom of a container. In this case, our result agrees to some recent experiments in which it is shown that the force exerted by the liquid is a downward force that increases linearly with depth, in clear contrast to the force predicted by AP. The method introduced here is indeed capable of providing a formula for the correct force, valid for bodies with arbitrary shapes, which involves the area of contact. Since Archimedes was one of the greatest geniuses of the ancient world, it would not be surprising that he had enunciated his theorems with remarkable precision and insight, however there are some instances he did not realize. These cases are shown here to be exceptions to the AP, thus it would be insensate to make great efforts to keep AP valid without exceptions at the cost of deficient redefinitions. Since the method presented here is not so complex mathematically, involving only basic rules of vector calculus, it could be included or mentioned in textbooks, at least in the form of a reference that could be looked up by the more interested readers. Appendix A: Proof of the gradient theorem Let us show how the Gauss’s divergence theorem (see, e.g., Refs. [10, 11, 22, 23]) can be applied to proof the gradient theorem. Divergence theorem. Suppose that R and S satisfy the conditions mentioned in the gradient theorem. If E = E(r) is a vector field whose components have continuous partial derivatives in all points of V (including S), then ∫∫ © S E · n̂ dS = ∫∫∫ R ∇ · E dV . Proof (of gradient theorem). Let P be a point over the closed surface S that bounds R. Suppose that f = f(r) has continuous partial derivatives at every point in R, including Checking the validity of Archimedes’ law 12 those at S. By choosing E = f c, c 6= 0 being an arbitrary constant vector, and substituting it in the integrals of the divergence theorem, above, one finds∫∫ © S (f c) · n̂ dS = ∫∫∫ R ∇ · (f c) dV . (25) Since ∇ · c = 0, then ∇ · (f c) = f (∇ · c) + c ·∇f = c ·∇f . Therefore ∫∫ © S c · (f n̂) dS = ∫∫∫ R c · (∇f) dV , which implies that c ·   ∫∫ © S f n̂ dS − ∫∫∫ R ∇f dV   = 0 . By hypothesis, c 6= 0. If the vector into parentheses, above, were not null, it should always be perpendicular to c, in order to nullify the scalar product, which is impossible because c is an arbitrary vector. Therefore, one has to conclude that∫∫ © S f n̂ dS = ∫∫∫ R ∇f dV . ✷ [1] T. L. Heath, The Works of Archimedes – Edited in modern notation (Cambridge Univ. Press, Cambridge, UK, 2010), pp. 253–262. [2] E. H. Graf, “Just What Did Archimedes Say About Buoyancy?,” Phys. Teacher 42, 296–299 (2004). [3] Note that Archimedes did not call his discoveries in hydrostatics by “laws,” nor did presented them as a consequence of experiments. Instead, he treated them as mathematical theorems, similarly to those proposed by Euclides for geometry. [4] D. Halliday, R. Resnick, and J. Walker, Fundamentals of Physics, 9th ed. (Wiley, New York, 2011), pp. 367–370. [5] R. Netz, “Proof, amazement, and the unexpected,” Science 298, 967–968 (2002). [6] D. C. Giancoli, Physics for Scientists and Engineers, 3rd ed. (Prentice Hall, New Jersey, 2000), pp. 340–343. [7] R. A. Serway and R. J. Beichner, Physics for Scientists and Engineers, 8th ed. (Brooks/Cole, Belmont, CA, 2010), pp. 408–412. [8] Of course, the only physically plausible cause for this upward force is the greater fluid pressure on the bottom of the cylinder in comparison to the pressure on the top. [9] The origin of the BF is not found in the Archimedes’ original work. [10] R. Courant and F. John, Introduction to calculus and analysis, vol. II (Springer, Berlim, 1999), pp. 590–607. [11] J. Stewart, Calculus - International Student Edition, 5th ed. (Thomson, Belmont, CA, 2003), pp. 1163–1169. [12] R. W. Fox, A. T. McDonald, and P. J. Pritchard, Introduction to Fluid Mechanics, 6th ed. (Wiley, New York, 2004), pp. 52–82. [13] G. E. Jones and W. P. Gordon, “Removing the buoyant force,” Phys. Teach. 17, 59–60 (1979); J. R. Ray and E. Johnson, “Removing the buoyant force, a follow-up,” Phys. Teach. 17, 392–393 (1979). [14] B. M. Valiyov and V. D. Yegorenkov, “Do fluids always push up objects immersed in them?,” Phys. Educ. 35, 284–285 (2000). Checking the validity of Archimedes’ law 15 ˆ 2 n S0 1n̂ g z 0 fluid 1 fluid 2 Figure 2. An arbitrarily-shaped body floating in a liquid (fluid 1), with the emerged part in contact to a less dense, compressible fluid (fluid 2). Note that n̂1 (n̂2) is the outward unit vector on the surface S1 (S2), as defined in the text. S0 is the horizontal cross-section at the level of the planar interface between fluids 1 and 2 (at z = 0). Checking the validity of Archimedes’ law 16 H F F Figure 3. The hydrostatic forces acting on rectangular blocks in contact to the walls of a container. The arrows indicate the pressure forces exerted by the liquid on the surface of each block. The net force exerted by the liquid in each block, i.e. the ‘buoyant’ force (as defined in the text), is represented by the vector F. The larger block represents the bottom case.