What is the buoyant force on a block at the bottom of a beaker
of water?
Carl E. Mungan
Physics Department, U.S.
Naval Academy
Annapolis, MD 21402-5002
Telephone: (410) 293-6680
Fax: (410) 293-3729
Email: mungan@usna.edu
Abstract
I propose that buoyant force be generally defined
as the negative of the total weight of the fluids that are displaced, rather
than as the net force exerted by fluid pressures on the surface of an object.
In the case of a body fully surrounded by fluids, these two definitions
are equivalent. However, if the object makes contact with a solid surface
(such as the bottom of a beaker of liquid), only the first, volumetric
definition is well defined while the second definition ambiguously depends
on how much fluid penetrates between the object and the solid surface.
Several recent papers [1-3]
have revived questions about the nature of the buoyant force on a submerged
object that is not fully surrounded by fluid. Suppose it makes contact with
a solid surface, such as a rectangular block firmly pressed to the bottom of
a beaker of water. An earlier pair of papers [4-5] suggests that in such a case the buoyant
force has been *removed*. Others argue that while a buoyant force still
exists, its direction is now *downward* [6]. The logic behind both of
these viewpoints is evident, but which one is consistent with introductory
physics textbooks? Open your favorite text and see if it answers this question.
You will probably find that it does not. Conventional books introduce buoyant
force by considering an object suspended in a liquid (perhaps by a string
of negligible cross-sectional area) so that it is fully surrounded by a single
fluid. Alternatively the body is floating and is thus surrounded by two fluids.
But the question of what happens when an object is only partly surrounded
by fluid is passed over in silence.
This silence leads to a nontrivial pedagogical
issue for introductory physics [7]. Consider drawing a free-body diagram
for a block on a table including the effects of the atmosphere. Should this
diagram include a buoyant force and, if so, in what direction [8]?
To resolve this ambiguity, we need to *clarify
the definition* of buoyant force [9]. Consider the following model situation.
A block of lower density than a fluid is held down at the bottom of a beaker
of the fluid as a result of the reduced pressure inside a suction cup [10]
(or thin o-ring) spanning the block's bottom face. The block has mass *m* and
top and bottom surfaces of area *A*, while the plastic of the suction
cup has negligible mass and volume. Define *P*_{top} to
be the fluid pressure at the depth of the top surface of the block, and
let *P*_{bottom,inside} and *P*_{bottom,outside} be
the fluid pressures at the depth of the bottom surface of the block respectively
inside and outside the volume of fluid enclosed by the suction cup. The
existence of suction implies that *P*_{bottom,inside} < *P*_{bottom,outside}. The weight of the block is balanced
by the difference in fluid forces on the bottom and top of the block and
by a normal force *N* exerted by the semi-rigid side walls of the suction cup,
*mg* = *N* + *AP*_{
bottom,inside} −
*AP*_{
top}. (1)
We can express the pressure inside the suction
cup as the difference between the pressure in the surrounding fluid at the
same depth and the pressure differential across the membrane of the suction
cup, *P*_{bottom,inside} = *P*_{bottom,outside} − D*P*.
This can be substituted into Eq. (1) to obtain
*mg* = *N* + *B* − *F*, (2)
where the magnitude of the buoyant force *B* has
here been* *defined to be equal to the weight of fluid displaced by
the block, and *F* = *A* D*P* is the "holding" force due to the
suction.
Although Eqs. (1) and (2) are fully equivalent and
both contain exactly one upward and one downward fluid force term, there
are three advantages of the second equation over the first:
1. We can use Eq. (2) to immediately compute
the minimum force *F*_{min} required to hold the block down, by setting *N* = 0.
One finds the intuitively appealing result that it is equal to the negative
of the block's apparent weight *mg* - *B*. In contrast, the hold-down
force is not explicit in Eq. (1).
2. We have separated the variation in fluid pressure with
depth from the pressure differential D*P* due
to the suction. This is a pedagogically instructive distinction to make.
3. Equation (2) consistently *defines* the
buoyant force on an immersed object to be upward and equal in magnitude to
the weight of fluid displaced, even when the object makes contact with solid
surfaces [11]. This definition remains simple and unambiguous if D*P* is nonzero.
Equation (2) also holds for a block (of arbitrary
density) on a table, if we broaden *F* to include not just the force
resulting from suction, but also from such effects as surface tension, cold
welding, and electrostatic surface charge interactions when appropriate [12].
But usually these effects are negligible. With that understanding, it is
reasonable to ask students, "What is the magnitude of the force required
to slowly lift the block?" A good first approximation is its weight *mg*.
If the problem asks us to account for the effects of the fluid environment
(such as the atmosphere) on an ordinary block, the correct answer would then
be its apparent weight *mg* − *B* [13]. It is only when there
is a reasonable expectation that the block is somehow coupled or sealed [14]
to the table that one needs to include additional forces *F*. This is
entirely analogous to how projectiles are treated in introductory physics:
One initially takes them to be in freefall. Subsequently a velocity-dependent
drag force can be added to account for air resistance. But additional effects
such as lift are only modeled under special circumstances.
# References
[1] J. Bierman and E. Kincanon, "Reconsidering
Archimedes' principle," *Phys. Teach.* **41**, 340-344 (2003).
[2] B.M. Valiyov and V.D. Yegorenkov, "Do fluids
always push up objects immersed in them?" *Phys. Educ.* **35**, 284-286
(2000).
[3] E.H. Graf, "Just what did Archimedes say
about buoyancy?" *Phys. Teach.* **42**, 296-299 (2004).
[4] G.E. Jones and W.P. Gordon, "Removing the
buoyant force," *Phys. Teach.* **17**, 59-60 (1979).
[5] J.R. Ray and E. Johnson, "Removing the buoyant
force: A follow-up," *Phys. Teach.* **17**, 392-393 (1979).
[6]
"Downward" and "upward" are relative to the direction of the effective gravitational
field **g** *including* the acceleration of the reference frame.
On a carousel, for example, "upward" is tilted toward the axis of rotation,
as shown in H.J. Haden, "A demonstration of Newtonian and Archimedean forces," *Phys.
Teach.* **2**, 176-177 (1964). As another example, on an elevator accelerating
downward, the weight of and buoyant force on a submerged object are decreased.
[7] J. Harper, "Archimedes' principle and the
FCI," *Phys. Teach.* **41**, 510 (2003).
[8] Place a block of mass *m* on the submerged
tray of a spring scale, as described in P.A. Tipler and G. Mosca, *Physics
for Scientists and Engineers*, 5th ed. (New York, Freeman, 2003), Sec.
13‑3. Tare the scale while the block is submerged but before it contacts
the tray. If you now place the block on the tray, the scale reading will
be *mg* − *B* where *B* is the weight of fluid displaced
by the block. This neither proves nor disproves that there is a buoyant force
on the block *alone*. For example, if fluid is squeezed out between
the block and tray, the downward fluid force on the block *increases* and
that on the tray *decreases* by exactly the same amount. That is, the
scale is actually sensitive to the buoyant force on the *combination* of
the block and tray, but its reading results from the manner of taring. (For
further discussion of methods of weighing a block in contact with the bottom
of a fluid, see Ref. 3.)
[9] The integral of the fluid pressure over
the surface of a body could be sideways (e.g., on a submerged block in contact
with the wall of an aquarium). Therefore taking that to be the general definition
of buoyant force would be misleading: A dictionary defines "buoyant" to mean
upward, as the reader is invited to check. In any case such an integral cannot
be evaluated if the extent of fluid seepage at the solid interface is unknown.
[10] Interestingly, the adhesion of many common
tapes results from suction, as discussed in R. Kunzig, "The physics of tape," *Discover* **20**,
27-29 (July 1999).
[11] Archimedes' principle is deduced in R.E.
Vermillion, "Derivations of Archimedes' principle," *Am. J. Phys.* **59**,
761-762 (1991) by considering the net change in the gravitational potential
energy of *both* the fluid and block during a virtual upward displacement
of the block. No fluid need initially be under the block for this derivation
to hold.
[12] For example, if fluid seepage under the
block can occur, so that suction cannot be sustained, then some other effect
such as a glue must exert force *F* to hold down a block whose density
is less than that of the fluid. This is consistent with my analysis in C.E.
Mungan, "Reprise of a 'Dense and tense story'," *Phys. Teach.* **42**,
292-294 (2004).
[13] The effect of atmospheric buoyancy on scale
readings is memorably illustrated in H. Stokes and W.D. Peterson, "Buoyancy
of air: An attention-grabbing demonstration," *Announcer* **33**,
94 (Summer 2003), as described on the web at <http://stokes.byu.edu/alkaseltzer.html>.
[14] A striking demonstration of an object sealing
to a surface is the "Atmospheric Pressure Demonstrator" currently sold by
PASCO. It consists of a rubber sheet (with a knob attached to its top) that
when slapped down onto a stool can be used to lift it. |