Using surface integrals for checking Archimedes' law of buoyancy

13 Pages·2012·0.21 MB·English

IOPPUBLISHING EUROPEANJOURNALOFPHYSICS
Eur.J.Phys.33(2012)101–113 doi:10.1088/0143-0807/33/1/009
Using surface integrals for checking
Archimedes’ law of buoyancy
F M S Lima
InstituteofPhysics,UniversityofBrasilia,POBox04455,70919-970Brasilia-DF,Brazil
E-mail:fabio@fis.unb.br
Received27July2011,infinalform5October2011
Published22November2011
Onlineatstacks.iop.org/EJP/33/101
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 the literature, which may be attributed to our trust in
Archimedes’lawofbuoyancy. However,thislaw,alsoknownasArchimedes’
principle(AP),doesnotyieldtheforceobservedwhenthebodyisincontact
withthecontainerwalls,asismoreevidentinthecaseofablockimmersedina
liquidandincontactwiththebottom,inwhichadownwardforcethatincreases
withdepthisobserved. Inthiswork,bytakingintoaccountthesurfaceintegral
ofthepressureforceexertedbyafluidoverthesurfaceofabody,thegeneral
validityofAPischecked. Forabodyfullysurroundedbyafluid,homogeneous
ornot,agradientversionofthedivergencetheoremapplies,yieldingavolume
integralthatsimplifiestoanupwardforcewhichagreeswiththeforcepredicted
by AP, as long as the fluid density is a continuous function of depth. For the
bottomcase,thisapproachyieldsadownwardforcethatincreaseswithdepth,
which contrasts to AP but is in agreement with experiments. It also yields a
formulaforthisforcewhichshowsthatitincreaseswiththeareaofcontact.
(Somefiguresinthisarticleareincolouronlyintheelectronicversion)
1. Introduction
ThequantitativestudyofhydrostaticphenomenabeganinantiquitywithArchimedes’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].1 In modern texts,
his propositions are reduced to a single statement known as Archimedes’ law of buoyancy,
orsimplyArchimedes’principle(AP),whichstatesthat‘anyobjectimmersedinafluidwill
experience an upward force equal to the weight of the fluid displaced by the body’ [3].2
1 Certainlyinconnectiontotheneed,atthattimeofimprovingshippingbysearoutes,ofpredictinghowmuch
additionalweightashipcouldsupportwithoutsinking.
2 NotethatArchimedesdidnotcallhisdiscoveriesinhydrostatics‘laws’,nordidhepresentthemasaconsequence
ofexperiments.Instead,hetreatedthemasmathematicaltheorems,similartothoseproposedbyEuclidforgeometry.
0143-0807/12/010101+13$33.00 c 2012IOPPublishingLtd PrintedintheUK&theUSA 101
!
102 FMSLima
The development of this work had to wait about 18 centuries, until the rise of the scientific
method, which was a guide for the experimental investigations in hydrostatics by Stevinus,
Galileo,Torricelli,andPascal,amongothers3. TheverylongtimeintervalfromArchimedes
totheseexperimentalistsisaclearindicationofhowadvancedhisthoughtswere. Aspointed
out by Netz (based upon a palimpsest discovered recently), Archimedes developed rigorous
mathematical proofs for most of his ideas [4]. However, the derivation of the exact force
exertedbyaninhomogeneousfluidonanarbitrarilyshapedbodyimmersedinit, aswillbe
shownhere,demandstheknowledgeofthedivergencetheorem,amathematicaltoolthatwas
outofreachfortheancients. Therefore,thevalidityoftheArchimedespropositionsforthis
moregeneralcasewasnotformallyprovedinhisoriginalwork(see[1]).
By defining the buoyant force (BF) as the net force exerted by a fluid on the portion of
thesurfaceofabody(fullyorpartiallysubmerged)thattouchesthefluid,thevalidityofAPin
predictingthisforcecanbechecked. Infact,thesimplecaseofsymmetricsolidbodies(e.g.,
aright-circularcylinder,asfoundin[5],orarectangularblock,asfoundin[6])immersedina
liquidisusedinmosttextbooksforprovingthevalidityofAP.There,symmetryargumentsare
takenintoaccounttoshowthatthehorizontalforcesexertedbytheliquidcancelandthenthe
netforcereducestothedifferenceofpressureforcesexertedonthetopandbottomsurfaces4.
The BF is then shown to point upwards, with a magnitude that agrees with AP, which also
explainstheoriginoftheBFintermsofanincreaseofpressurewithdepth5. Note,however,
thatthisproofworksonlyforsymmetricbodieswithhorizontal,flatsurfacesonthetopandthe
bottom, immersedinanincompressible(i.e.homogeneous)fluid. Althoughanextensionof
thisresultforarbitrarilyshapedbodiesimmersedinaliquidcanbefoundinsometextbooks
[7–9], a formal generalization for bodies immersed in a compressible (i.e. inhomogeneous)
fluidisnotfoundintheliterature. Thiscertainlyinducesthereaderstobelievethatitshould
beverycomplexmathematically,whichisnottrue,aswillbeshownhere.
The existence of exceptions to Archimedes’ law has been observed in some simple
experimentsinwhichtheforcepredictedbyAPisqualitativelyincorrectforabodyimmersed
inafluidandincontactwiththecontainerwalls. Forinstance,whenasymmetricsolid(e.g.,
a cylinder) is fully submerged in a liquid with a face touching the bottom of a container, a
downward BF is observed, as long as no liquid seeps under the block [2, 10, 11, 12, 14].
Indeed,theexperimentalevidencethatthisforceincreaseswithdepth(see,e.g.,[2,12–14])
clearlycontraststotheconstantforcepredictedbyAP.Thesedisagreementsledsomeauthors
toreconsiderthecompletenessorcorrectnessoftheAPstatement,aswellasthedefinitionof
BFitself[2,11–13,15],whichseemstomakethethingsyetmoreconfusing.
On aiming at checking mathematically the validity of AP for arbitrarily shaped bodies
immersedininhomogeneousfluidsandintendingtoelucidatewhythebottomcaserepresents
anexception toAP,inthisworkImake useofsurfaceintegrals forderiving theexact force
exerted by a fluid on the surface of a body immersed in it. When the body does not touch
thecontainerwalls, agradientversionofthedivergencetheoremapplies, yieldingavolume
integralthatreducestotheweightofthedisplacedfluid,inagreementwithAP.Forthebottom
case, this approach yields a force that points downwards and increases with depth, in clear
disagreementwithAP,butinagreementwithexperiments. Thismethodalsorevealsthatthis
3 Gravesandealsodeservescitation,duetohisaccurateexperimentforcomparingtheforceexertedbyaliquidon
abodyimmersedinittotheweightofthedisplacedliquid. Thisexperimentusesabucketandametalliccylinder
thatfitssnuglyinsidethebucket. Bysuspendingthebucketandthecylinderfromabalanceandbringingitinto
equilibrium,oneimmersesthecylinderinawatercontainer. Thebalanceequilibriumisthenrestoredbyfillingthe
bucketwithwater.
4 Ofcourse,theonlyphysicallyplausiblecauseforthisupwardforceisthegreaterfluidpressureonthebottomof
thecylinderincomparisontothepressureonthetop.
5 TheoriginoftheBFisnotfoundinArchimedes’originalwork[1].
CheckingthevalidityofArchimedes’law 103
Figure1. AcylinderofheighthandradiusR,fullyimmersedinaliquid. NotethattheBFB
isverticalanddirectedupwards. ThetopsurfaceS2iskeptatadepthz2 belowtheliquid–air
interface(aplaneatz 0).
=
forcedependsontheareaofcontactA betweenthebodyandthebottom,beingequaltothe
b
difference between the product p A (p being the pressure at the bottom) and the weight
b b b
of the displaced fluid, a result that is not found in the literature and could be explored in
undergraduateclasses.
2. Buoyantforceonabodywitharbitraryshape
In modern texts, Archimedes’ original propositions are reduced to a single, short statement
knownasArchimedes’principle[3,6].
When a body is fullyor partially submerged ina fluid, a buoyant force B from the
surroundingfluidactsonthebody. Thisforceisdirectedupwardandhasamagnitude
equaltotheweightm gofthefluiddisplacedbythebody.
f
Here,m isthemassofthefluidthatisdisplacedbythebodyandgisthelocalacceleration
f
ofgravity. TheBFthatfollowsfromAPissimply
B m g, (1)
f
=−
where g gkˆ, kˆ being the unit vector pointing in the z-axis direction, as indicated in
=−
figure 1. For an incompressible, homogeneous fluid—i.e. a fluid with a nearly uniform
density6—,onehasm ρV ,andthen
f f
=
B ρV gkˆ, (2)
f
=
whereρ isthedensityofthefluidandV isthevolumeofthefluidcorrespondingtom .7 In
f f
mosttextbooks,someheuristicargumentssimilartotheStevinus‘principleofrigidification’
(see,e.g.,[16])aretakenintoaccountforextendingthevalidityofthisformulatoarbitrarily
6 Forsmalldepths,mostliquidsbehaveaspracticallyincompressiblefluids. Forinstance,thedensityofseawater
increasesbyonly0.5%whenthe(absolute)pressureincreasesfrom1to100atm(atadepthof1000m).
7 Ofcourse,whenthebodyisfullysubmerged,Vf equalsthevolumeVofthebody.
104 FMSLima
shapedbodiesimmersedinaliquid(see,e.g.,[3,5,6]),withoutamathematicaljustification.
Thevalidityofthisgeneralizationischeckedbelow.
Our method starts from the basic relation between the pressure gradient in a fluid in
equilibrium in a gravitational field g and its density ρ ρ(r), with r being the position
=
vector. Fromtheforcebalanceforanelementofvolumeofthefluid(homogeneousornot)in
equilibrium,onefindsthewell-knownhydrostaticequation[9,17]
p ρ(r)g. (3)
∇ =
For a uniform, vertical (downward) gravitational field, as will be assumed hereafter8, this
simplifiesto
∂p
p kˆ ρ(z)g kˆ. (4)
∇ = ∂z =−
Forahomogeneous(i.e.incompressible)fluid, ρ isaconstant(seefootnote6),andthen
(4)canbereadilyintegrated,yielding
p(z) p ρgz, (5)
0
= −
wherep isthepressureatz 0,anarbitraryreferencelevel. Thislineardecreaseofpressure
0
=
withheightisknownastheStevinuslaw[3,5,6].
LetusnowderiveageneralformulaforBFevaluations,i.e.onethatworksforarbitrarily
shapedbodies,fullyorpartiallysubmergedinafluid(orsetofdistinctfluids)homogeneous
ornot. Toavoidconfusion,letusdefinetheBFasthenetforcethatafluidexertsonthepartS
ofthesurface#ofabodythatiseffectivelyincontactwiththefluid. Ofcourse,ifthebodyis
fullysubmerged,thesurfaceScoincideswith#. Consider,then,abodyimmersedinafluid
instaticequilibrium. Sincethereisnoshearingstressinafluidatrest,thedifferentialelement
offorcedFitexertsonadifferentialelementofsurfacedS,atapointPofSinwhichthefluid
touches(andpushes)thebody,isnormaltoSbyP.Therefore,
dF p(r)dS nˆ, (6)
=−
where nˆ nˆ(r) is the outward normal unit vector at point P. Note that the pressure force
=
is directed along the inward normal to S by P. On assuming that the surface S is piecewise
smooth and the vector field p(r)nˆ is integrable over S, one finds a general formula for BF
evaluations,namely
F p nˆ dS. (7)
=−
!!S
Thisintegralcanbeeasilyevaluatedforabodywithasymmetricsurfacefullysubmergedin
afluidwithuniformdensity. Foranarbitrarilyshapedbody,however,itdoesnotappeartobe
tractableanalyticallyduetothedependenceofthedirectionofnˆonthepositionroverS,which
inturndependsonthe(arbitrary)shapeofS.However,thistaskcanbeeasilyworkedoutfor
a body fully submerged in a homogeneous fluid (ρ constant) by applying the divergence
theorem to the vector field E p(z)kˆ, which yie=lds a volume integral that evaluates to
ρgV kˆ, in agreement with AP=, a−s discussed in some advanced texts [7, 20]. For the more
general case of a body fully or partially submerged in an inhomogeneous fluid (or a set of
fluids),Ishallfollowaslightlydifferentroutehere,baseduponthefollowingversionofthe
divergencetheorem[7,19].9
8 TheassumptionofauniformgravitationalfieldisaverygoodapproximationforpointsneartheEarth’ssurface.
However, forproblemsinvolvinglargedepths/heightsoffluids(e.g., underseaextractionofoil, isostasyindeep
crustallayers,andothers),thedecayofthegravitationalfieldwiththedistancetothecentreoftheEarthhastobe
takenintoaccount. Fortheseapplications,amoreaccuratemodel(reconsideringequations(4)and(5))hastobe
developed.
9 Thismoregeneralversionofthedivergencetheoremisusedherebecauseitisvalidforpiecewisesmoothsurfaces,
thuscovering,e.g.,thecasesofrectangularblocksandcylinders,inwhichSisnotsmoothinallitspoints.
CheckingthevalidityofArchimedes’law 105
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 nˆ directed
outwardsfromR.Iff f(r)isascalarfunctionwithcontinuouspartialderivativesinall
=
pointsofanopenregionthatcontainsR(includingS),then
f(r)nˆ dS f dV.
" = ∇
!! !!!R
S
In the appendix, it is shown how the divergence theorem can be used for proving the
gradienttheorem. Theadvantageofusingthisless-knowncalculustheoremisthatitallows
ustotransformthesurfaceintegralinequation(7)intoavolumeintegralof p,avectorthat
∇
canbeeasilywrittenintermsofthefluiddensityviathehydrostaticequation(4). Forthis,let
ussubstitutef(r) p(r)intobothintegralsofthegradienttheorem. Thisyields
=−
p(r)nˆ dS pdV , (8)
− " =− ∇
!! !!!Vf
S
where the surface integral on 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
analysethismoreclosely.
2.1.Abodyfullysubmergedinafluid
Foranarbitraryshapebodyfullysubmergedinafluid(orasetoffluids),bysubstitutingthe
pressuregradientofequation(4)intoequation(8),onefinds10
F pdV ρ(z) dV gkˆ. (9)
=− !!!Vf ∇ ="!!!Vf #
For the general case of an inhomogeneous, compressible fluid whose density changes
with depth, as occurs with gases and high columns of liquids11, the pressure gradient in
equation (9) will be integrable over V as long as ρ(z) is a continuous function of depth (in
f
conformitytothehypothesisofthegradienttheorem). Withthiscondition,onehas
F ρ(z) dV gkˆ ρ(z) dV gkˆ. (10)
="!!!Vf # =$!!!V %
Since ρ(z) dV isthemass m offluidthatwouldoccupythevolumeVofthebody(fully
V f
submerged),
&&&
F m g kˆ, (11)
f
=
which is an upward force whose magnitude equals the weight of the fluid displaced by the
body, in agreement with AP as stated in equation (1). This shows that AP remains valid
evenforaninhomogeneousfluid, aslongasthedensityisacontinuousfunctionofdepth, a
conditionfulfilledinmostpracticalsituations.
10For a homogeneous fluid, one has ρ(r) ρ const, which reduces the integral in equation (9) to
= =
ρg Vf dV kˆ =ρgVfkˆ =ρgV kˆ,which,asexpected,agreeswithAP.
11B’e&i&n&grigoro(us,allfluidspresentsomeincreaseofdensitywithdepth,buttheincreasingrateforliquidsismuch
smallerthanthatforgases.
106 FMSLima
2.2.Abodypartiallysubmergedinafluid
The case of an arbitrarily shaped body floating in a fluid with a density ρ (z), with its non-
1
submerged part exposed to either vacuum (i.e. a fictitious fluid with null density) or a less
densefluid,isaninterestingexampleoffloatinginwhichtheexactBFcanbecomparedtothe
forcepredictedbyAP12. Thisisimportantforthestudyofmanyfloatingphenomena, from
shipsinseawatertotheisostaticequilibriumoftectonicplates(knowningeologyasisostasy)
[28]. Without loss of generality, let us restrict our analysis to two fluids: one (denser) with
densityρ1(z),wecallfluid1;another(lessdense)withdensityρ2(z)!min[ρ1(z)] ρ1(0−),
=
wecallfluid2. Forsimplicity, Ichoosetheorigin z 0at theplanar surfaceof separation
=
betweenthefluids,asindicatedinfigure2,wherethefluiddensityusuallypresentsa(finite)
discontinuity ρ (0 ) ρ (0+). Theforcesthatthesefluidsexertonthebodysurfacecanbe
1 − 2
−
evaluatedbyapplyingthegradienttheoremtoeachfluidseparately,asfollows. First,divide
the body surface S into two parts: the open surface S below the interface at z 0 and the
1
=
opensurfaceS abovez 0. Theintegraloverthe(closed)surfaceSinequation(7)canthen
2
=
bewrittenas
p(z)nˆ dS p(z)nˆ dS+ p(z)nˆ dS, (12)
1 2
" =
!! !!S1 !!S2
S
wherenˆ (nˆ )istheoutwardunitnormalvectoratapointofS (S ),asindicatedinfigure2.
1 2 1 2
Let us call S the planar surface, also indicated in figure 2, corresponding to the horizontal
0
cross-sectionofthebodyatz 0. Bynotingthatnˆ kˆ andnˆ kˆ inallpointsofS ,
1 2 0
= = =−
p(z)beingacontinuousfunction,onehas
p(z)nˆ dS+ p(z)nˆ dS 0.
1 2
=
!!S0 !!S0
This allows us to use S to generate two closed surfaces, S and S , formed by the unions
0 1 2
S S andS S ,respectively. Fromequation(12),onehas
1 0 2 0
∪ ∪
) )
p(z)nˆ dS p(z)nˆ dS+ p(z)nˆ dS+ p(z)nˆ dS+ p(z)nˆ dS
1 1 2 2
" =
!! !!S1 !!S0 !!S2 !!S0
S
p(z)nˆ dS+ p(z)nˆ dS. (13)
1 2
= " "
!! !!
S1 S2
As both S1 and S2)are closed surfa)ces, one can apply the gradient theorem to each of them,
separately. Thisgives
) )
F p(z)nˆ dS+ p(z)nˆ dS
1 2
=− " "
*!! !! +
S1 S2
) pdV + ) pdV (14)
=− ∇ ∇
*!!!V1 !!!V2 +
∂p ∂p
dV + dV kˆ,
=− ∂z ∂z
*!!!V1 !!!V2 +
whereV andV arethevolumesoftheportionsofthebodybelowandabovetheinterfaceat
1 2
z 0,respectively. Fromthehydrostaticequation,onehas ∂p/∂z gρ(z) ,whichreduces
= =−
12AnullpressureisassumedontheportionsofSthatarenotinteractingwithanyfluid.
CheckingthevalidityofArchimedes’law 107
Figure2.Anarbitrarilyshapedbodyfloatinginaliquid(fluid1),withthenon-submergedpartin
contactwithalessdense,compressiblefluid(fluid2).Notethatnˆ1(nˆ2)istheoutwardunitvector
onthesurfaceS1(S2),asdefinedinthetext. S0isthehorizontalcross-sectionatthelevelofthe
planarinterfacebetweenfluids1and2(atz 0).
=
theaboveintegralsto
[ ρ (z)g] dV + [ ρ (z)g] dV
1 2
− −
!!!V1 !!!V2
g ρ (z)dV + ρ (z)dV . (15)
1 2
=−
$!!!V1 !!!V2 %
Thelattervolumeintegralsareequivalenttothemassesm andm offluids1and2displaced
1 2
bythebody,respectively,whichreducestheBFto
F g ρ (z)dV + ρ (z)dV kˆ (m +m ) gkˆ. (16)
1 2 1 2
= =
$!!!V1 !!!V2 %
The BF is then upwards and its magnitude is equal to the sum of the weights of the fluids
displaced by the body, in agreement with AP in the form given in equation (1). Note
that the potential energy minimization technique described in [21–23] cannot provide this
confirmation of AP because it works only for rigorously homogeneous (i.e. incompressible)
fluids. Interestingly,ourproofshowsthattheexactBFcanalsobefoundbyassumingthatthe
bodyisfullysubmergedinasinglefluidwithavariabledensityρ(z) thatisnotcontinuous,
butapiecewisecontinuousfunctionwitha(finite)leapdiscontinuityatz 0.13
=
Although the contribution of fluid 2 to the BF is usually smaller than that of fluid 1, it
cannot in general be neglected, as has been done in introductory physics textbooks [24]. In
ourapproach,thiscorrespondstoassumingaconstantpressureonallpointsofthesurfaceS
2
of the emerged portion, which is incorrect. This leads to a null gradient of pressure on the
non-submergedpartofthebody,whicherroneouslyreducestheBFtoonly
F ρ (z)dV gkˆ. (17)
1
=
$!!! %
V1
Fluid1beingaliquid,asusual,ρ (z)isnearlyaconstant(letuscallitρ ),whichsimplifiesthis
1 1
upwardforcetoρ V g. Thisistheresultpresentedforthewater–airpairinmosttextbooks
1 1
13Interestingly,thissuggeststhatamoregeneralversionofthedivergencetheorem(seetheappendix)couldbefound,
inwhichtherequirementofcontinuityofthecomponentsofE E(r)isweakenedtoonlyapiecewisecontinuity.I
=
havenotfoundsuchageneralizationintheliterature.
108 FMSLima
[3,5,6]. ItisalsotheresultthatcanbededucedfromArchimedes’originalpropositions14,as
wellastheresultfoundbyminimizingthepotentialenergy[21–23]. Itisclearthatthisnaive
approximateresultalwaysunderestimatestheactualBFestablishedinequation(16),beinga
reasonableapproximationonlywhenm m .15Theinclusionofthetermcorrespondingto
2 1
$
m isthenessentialforaccurateevaluationsoftheBF,asshownbyLanforablockfloating
2
in a liquid [24]. For an arbitrarily shaped body immersed in a liquid–gas fluid system, our
equation(15)yieldsthefollowingexpressionfortheexactBF:
∂p
F ρ gV dV kˆ. (18)
1 1
= − ∂z
* !!!V2 +
Therefore,thefunctionp(z)ontheemergedportionofthebodydeterminesthecontribution
of fluid 2 to the exact BF. If fluid 2 is approximated as an incompressible fluid, i.e. if one
assumesρ (z) ρ (0) ρ ,byapplyingtheStevinuslaw,onefinds
2 2 2
≈ ≡
F ρ gV [ ρ (0)g] dV kˆ (ρ gV +ρ gV )kˆ. (19)
1 1 2 1 1 2 2
≈ − − =
, !!!V2 -
ThisisjusttheresultfoundbyLanbyapplyingAPintheformstatedinourequation(2)to
fluids1and2,separately,andthensumminguptheresults[24]. Ofcourse,thisapproximation
isbetterthanequation(17),but,contrarilytoLan’sopinion,thecorrectresultarisesonlywhen
onetakesintoaccountthedecreaseofρ withz. Thisdemandstheknowledgeofabarometric
2
law,i.e.aformulaforp(z)withz>0. Fortunately,mostknownbarometriclawsarederived
justfromthehydrostaticequation[17],whichallowsustosubstitutethepressurederivative
intoequation(18)by ρ (z)g,findingthat
2
−
F ρ gV [ ρ (z)g]dV kˆ
1 1 2
= − −
$ !!!V2 % (20)
ρ V + ρ (z)dV g kˆ,
1 1 2
=
$ !!!V2 %
whichpromptlyreducestotheexactresultfoundinequation(16).16
3. ExceptionstoArchimedes’principle
Whenabodyisimmersedinaliquidandputincontactwiththecontainerwalls,asillustrated
infigure3, aBFisobservedwhichdoesnotagreewiththatpredictedbyAP.Thishasbeen
observed in some simple experiments in which a symmetric solid (typically a cylinder or
a rectangular block) is fully submerged in a liquid with a face touching the bottom of a
container. Let us call this the bottom case [10]. In this case, a downward BF (according to
our definition) is observed if no liquid seeps under the block [10–12, 14]. There is indeed
experimental evidence that the magnitude of this force increases linearly with depth (see,
e.g.,[2,12,14]),whichcontrastswiththeconstantforce( ρVg kˆ)predictedbyAP.These
=
experimentalresultshaveledsomeauthorstoreconsiderthecompletenessorcorrectnessof
theAPstatement,aswellasthedefinitionofBFitself[2,11–13,15].
14Archimedes certainly knew the existence of air, which was discovered by Empedocles (490–430BC), but, as
pointedoutbyRorres,itisdoubtfulthathewasawareofthebuoyancyeffectsofair.Seenote4in[25].
15Infact,formostapplicationsinvolvingaliquid–airpair,thesecondterminequation(19)representsonlyasmall
correctiontotheapproximationinequation(17)becauseairdensityisgenerallymuchsmallerthantypicalliquid
densities. Forinstance,atsealevel,theairdensityisonly1.2kgm 3,whereasthefreshwaterdensityisabout
−
1000kgm 3.
−
16When fluid 2 is vacuum—i.e. the body is partially submerged (literally) in fluid 1 only—the exact result in
equation(20), above, isindeedcapableoffurnishingthecorrectforce(namelyanupwardforcewithmagnitude
ρ1V1g),whichisfoundbytakingthelimitasρ2(z)tendstozerouniformly.
CheckingthevalidityofArchimedes’law 109
Figure3. The hydrostatic forces acting on rectangular blocks in contact with the walls of a
container. Thearrowsindicatethepressureforcesexertedbytheliquidonthesurfaceofeach
block.Thenetforceexertedbytheliquidineachblock,i.e.the‘buoyant’force(asdefinedinthe
text),isrepresentedbythevectorF.Thelargerblockrepresentsthebottomcase.
InspiteoftheexperimentaldifficultiesinstudyingtheseexceptionstoAP,Ishallapply
the surface integral approach to determine the exact BF that should be observed in an ideal
experimentinwhichthereisnofluidundertheblock. Thiswillhelpustounderstandwhythe
bottomcaserepresentsanexceptiontoAP.Forasymmetricsolidbody(eitheracylinderora
rectangularblock)withitsflatbottomrestingonthebottomofacontainer,asillustratedfor
theblockattherightoffigure3,ifnoliquidseepsundertheblock,thenthehorizontalforces
cancelandthenetforceexertedbytheliquidreducestothedownwardpressureforceexerted
bytheliquidonthetopsurface[10–12]. FromtheStevinuslaw,p p +ρg z ,z
top 0 top top
=
beingthedepthofthetop;therefore,
. .
. .
F p(z)nˆ dS p Akˆ (p +ρg z )Akˆ, (21)
top 0 top
=− =− =− | |
!!Stop
where A is the area of the top surface. This downward force then increases linearly with
depth, which clearly contrasts with the force predicted by AP, but agrees with experimental
results[2,12,14]. Infact,theincreaseofthisforcewithdepthhasbeenthesubjectofdeeper
discussions in recent works [2, 12, 13, 26], in which it is suggested that the meaning of the
word‘immersed’shouldbe‘fullysurroundedbyaliquid’insteadof‘incontactwithaliquid’,
which would make the ‘bottom’ case, as well as all other ‘contact cases’, out of the scope
ofArchimedes’originalpropositions,aswellasthemodernAPstatement[2,12,26]. Note,
however,thatthisredefinitionisdeficientbecauseitexcludessomecommoncasesofbuoyancy
such as, for instance, that of a solid (e.g., a piece of cork) floating in a denser liquid (e.g.,
water). Inthissimpleexample,thebodyisnotfullysurroundedbyaliquidandyetAPworks!
Morerecently,otherauthorshavearguedthatthedefinitionofBFitselfshouldbechangedto
‘anupwardforcewithamagnitudeequaltotheweightofthedisplacedfluid’[13]. However,
I have noted that this would make AP a definition for the BF and then, logically, AP would
notadmitanyexceptionsatall. FacedwiththedownwardBFexperimentsalreadymentioned,
itisclearthatthisisnotagoodchoiceofdefinition. Therefore, Iwouldliketoproposethe
abandonmentofsuchredefinitions, astheyareunnecessaryonceweadmitsomeexceptions
toAP,whichisthenaturalwaytotreattheexceptionalcasesnotrealizedbyArchimedesin
hisoriginalwork.
110 FMSLima
For a better comparison to the result for arbitrarily shaped bodies that will be derived
below,letuswritetheforcefoundinequation(21)intermsofthepressurep atthebottomof
b
thecontainer. Asthereadercaneasilycheck,thisyields
F (p A ρVg) kˆ. (22)
b
=− −
Note that this simple result is for a ‘vacuum’ contact, i.e. an ideal contact in which neither
liquidnorairisundertheblock. Thisisanimportantpointtobetakenintoaccountbythose
interestedindevelopingadownwardBFexperimentsimilartothatproposedbyBiermanand
Kincanon[12],sinceapartofthebottomoftheblockwithanareaA isintentionallyleftin
air
contactwithair(thiscomesfromtheirtechniquetoreducetheliquidseepageundertheblock)
[14]. ThischangestheBFto
F (p A p A ρVg) kˆ. (23)
b 0 air
=− − −
Foranarbitrarilyshapedbodyimmersedinaliquid(inthebottomcase),letusassume
thatthereisanon-nullareaA ofdirectcontactbetweenthebodyandthebottomofacontainer.
b
Ifnoliquidseepsundertheblock,thenthepressureexertedbytheliquidthereatthebottom
ofthebodyisofcoursenull. TheBFisthen
F p(z)nˆ dS p(z)nˆ dS+ 0( kˆ)dS
=− =− −
!!S2∪Ab $!!S2 !!Ab %
p(z)nˆ dS.
=−
!!S2
Inviewofapplyingthegradienttheorem,oneneedsasurfaceintegraloveraclosedsurface.
By creating a fictitious closed surface # S A on which the pressure forces will be
2 b
= ∪
exerted as if the body would be fully surrounded by the liquid (i.e. one assumes a constant
pressurep overthehorizontalsurfaceA ),onehas
b b
F p(z)nˆ dS p ( kˆ)dS+ p ( kˆ)dS
b b
=− − − −
!!S2 !!Ab !!Ab
p(z)nˆ dS p ( kˆ)dS p(z)nˆ dS p kˆ dS (24)
b b
=− " − − =− " −
!! !!Ab !! !!Ab
# #
ρVg kˆ p A kˆ (p A ρVg) kˆ.
b b b b
= − =− −
This is again a downward BF that increases linearly with depth, since p p +ρgH , H
b 0
=
beingtheheightoftheliquidcolumnabovethebottom,asindicatedinfigure3. Thisresultfor
arbitrarilyshapedbodiesisnotfoundintheliterature. Incidentally,thisresultsuggeststhatthe
onlyexceptionstoAP,forfluidsinequilibrium,arethosecasesinwhich pisnotapiecewise
∇
continuousfunctionoverthewholesurface# ofthebody;otherwisetheresultsoftheprevious
sectionguaranteethattheBFpointsupwardsandhasamagnitude ρVg,inagreementwith
AP.Thisincludesallcontactcases,sincetheboundaryofthecontactregioniscomposedof
pointsaroundwhichthepressureleaps(i.e.changesdiscontinuously)fromastrictlypositive
value p(z) (on the liquid side) to a smaller (ideally null) pressure (at the contact surface).