Buoyancy Measurements for Teaching and Research Bernard Mlller Textile Research Institute, Princeton. NJ 06542
-
One of the most useful and satisfvine .. accomolishments of an experimental scientist is thedesign uf'an experiment that can disorove a theorv. This is usuallv called a "fals~fication" test, because i t ma; prove that the theory is wrong but i t cannot prove that it is right. Indeed, no experimental evidence can guarantee that a theory is absolutely correct. A more positive use of such a test would be to eliminate one possible explanation for a phenomenon in favor of another. The latter would gain in credihility a t the expense of the former's demise. The following examination of the venerable buoyancy principle can serve as a simple but rigorous illustration of a falsification test that not only clears up a possible misconception but also points the way to a number of practical uses of huoyancy measurements that have not generally been recognized. The Buoyancy Prlnclple Most reference books describe the Archimedes buoyancy principle in the following n'ay: "A body submerged wholly r' partially inalluid is buoyed up by a forceequal to the weight of fluid d i i ~ l a c r dbv the hudv." When the disulacrd fluid completely bverflo& its continer, as i t did in Akhimedes's bathtub. that amount of liouid is easilv identified. Nor is there any problem when the budy is completely submerged. In both cases. the displaced volume is eaual to the volume of solid that is in the l&uid. However, when the solid is only partially submerged and there is no overflow from the liquid container, experience has shown that the word "displaced" can he construed in two ways. Suppose that the body is a cylindrical rod held by an external clamp with its axis vertical so that i t is partially submerged in liquid, as shown in Figure 1. For such a case, what is the volume of displaced liquid? One answer is that it is equal to the entire volume of solid that is beneath the surface of the liquid. On the other hand, others interpret "displaced liquid" to mean the amount of liquid that has been moved from its original location. In Figure 1,the latter would correspond to a ring of liquid represented by the crosshatched region. If these two volumes were equivalent, then the above semantic distinction is not important. However, if they are not, only one can be correctly applied to predict buoyancy force. If we start with the obvious relationship
BALANCE
I
Figure 1. Buoyancy farce experiment: a = cross-wctional area of rod; A = cross-sectional area of container: C =external clamp; I = depth of immersion of rod: & = initial height of liquid: AH= change in height of liquid.
I t is clear that the two volumes are not the same. Thenext question to be resolved is which is the correct one for predicting the generated huoyancy force. If we combine eqs 2 and 3 so as to eliminate AH, we get
This leaves us with two contradictory predictions. The buoyancy force FB is equal either to F,
= a . 1.6,
(6, = density of the liquid)
(5)
A Falslflcatlon Test
Volume of liquid Volume of Volume of +volume of immersed = liquid + immersed solid (constant) solid in terms of the dimensions shown in Figure 1,this gives us where A and a are the cross-sectional areas of the container and the rod, respectively. The volume of immersed solid AVs is therefore
The volume of liquid that has been moved AVL is equal to AV,=(A-n).AH
1
(3)
We can now state a simple verifiable distinction between the two oossibilities: Eauation 5 predicts that the slope of a plot of buoyancy force F B versus1 will not be dependent on A; equation 6 says that it will. Therefore, the falsification experiment becomes a simple matter of lowering a rod in unit immersion steps into a liquid in each of two containers that have different diameters and measuring the buoyancy force change resulting from each step. If the FBversus 1plots are found to have the same slope in both cases, the second interpretation would be invalid. The stepwise immersion-buoyancy force data needed for this test can be obtained without concern for the absolute values of 1. The rod can be started out a t any initial unrecorded immersion depth; all that is needed is M B / A ~the , Volume 66
Number 3 March 1969
267
Millimeter scale covered with transparent taps
The Wettlng Force If the stepwise immersion experiment is carried out with sufficient accuracy and sensitivity, i t may be possible to detect that the extrapolated linear plot does not go through the zerwzero point. Instead a small negative y intercept will be observed. This is due to the contribution of an additional force (of attraction) acting between the rod and the liquid, called the wetting force. Part of this pull on the liquid, which we designate as F,, is also sensed by the balance, so that the complete weight change produced by immersing the rod is actually (7)
AW=FB+Fw
Weight
For any given combination of rod material, liquid, and direction of rod movement, F, will be constant and independent of the devth of immersion. Therefore. this additional pull, F,, is thLintercept. In most cases F, will be negative, which simnlv means that the rod is vulling-up. on the liquid. However, F, can also he positive; which implies that there are cases where the rod is ~ u l l i n gdown on the liquid. This apparent contradiction expGined by the ~ i l h e l m ywetting force principle illustrated in Figure 4. The wetting force is the result of attraction between the surface of the liquid and its perimeter of contact on the surface of the rod, P. When this attraction is relatively great, the angle 0 between the solid surface and the liquid surface, called the contact &
Figure 2. Arrangement for buoyancy measurements with a triple-beam balance.
change'in buoyancy per unit change in 1, as long as the crosssectional area of the rod and the density of the liquid do not change. Measurlng Buoyancy For the purposes of these experiments, the best way to measure the buoyancy force is to place the container holding the liauid on a ton-loadine balance. As the rod is inserted. the haiance will &ow an iicrease in weight that is equal td the huovancv force. Looked a t in the context of Newton's third law, this should not be surprising; if the liquid pushes up on the rod, the latter must push back with equal force, and this downward force is transmitted to the balance. (For an alternate explanation of the downward force, see the Appendix.) Adequate data for the falsification test can be obtained using simple laboratory equipment and procedures such as the following:
-
Weight Change with lmmerslon Depth
Weight change
lmmnion
6.5 .cro-aiona~
6.5
44.1
42.6
areas: beaker A = 28.3 cm? beaker 6 = 50.3 c d .
(1) A glass, metal, or plastic rod about 15 em long and at least 3 em in diameter is provided with a length scale by taping a narrow strip of graph paper (reading to 1 mm) to it with waterproof
transparent tape. The tape should cover as little as possible of the surface of the rod. (2) The rod is held in a clamp attached to a stand and placed above a beaker of water that is on the weighing pan of a triple-heam balance reading to 0.1 g, as shown in Figure 2. (3) The balance is zeroed by adding weights to balance out the weight of the beaker plus water. (4) The rod is lowered until about 1 cm of it is submerged. The balance pointer will rise indicating increased loading on the balance. (5) Additional weights are added to bring the pointer back to its zero position. The depth of immersion and the amount ofweight added are recorded. (6)The rod is lowered another increment and step 5 repeated. The same is done for at least five immersion steps. (7) A plot is made of generated force (i.e., weight gain) against depth of immersion. (8)The experiment is repeated using a beaker that is significantly larger in diameter than the first one. Results for such a pair of experiments are shown in the table and Figure 3 for a hrass rod (diameter = 3 em) with two different-size beakers. I t is obvious from Figure 3 that the same slope was obtained with both beakers, and, therefore, eq 6 is not correct. Equation 5, based on the premise that the entire immersed volume of solid generates the huoyancy force, is sustained. 268
Journal of Chemical Education
WEIGHT CHANGE, g
0
rn
IMMERSION D E P M
BeakerA Beaker B
e . cm
Figure 3. Results lor immenlon of a brass rcd In two dinerent-sired beakers.
-
PULL ON LIQUID Fw=yPcose
L 7
\
.-+ O'i
Liquid
-
-
Figure 4. Wilhelmy wetling twce. The pull on the liqllid can be in either direction: y = surtace tension of llquld: P = perimeter of solid: 8 = contact angle.
tive force between it and water is considerably less than that for brass. An Appllcatlon of the Buoyancy Force Measurement A convenient application for this kind of buoyancy measurement is to determine the extent to which a liquid has penetrated into the available free volume within a porous network, or, in other words, how much air remains trapped within the material after it is immersed in a liquid. A simple illustrative experiment can be performed with a piece of fabric rolled up tightly in the form of a cylinder and held together with one or more paper clips (see Fig. 6). The fahric is pushed completely under the water by the wire holder (made from a coat haneer). and the weieht eain determined after, say, 1min. A blank'& with jus
