2008 AAPT Summer Meeting
Durham, NC 27708
I have used the resource “Peer Instruction: A User’s Manual” by Eric Mazur (Prentice Hall, Upper Saddle River, New Jersey, 1997) in my introductory physics courses almost since its appearance eleven years ago. I have presented many of these thought-provoking, stimulating, and downright mysterious problems to thousands of students via quizzes, exams, and PRS/Interactive Lecture Demonstration type questions in myriad forms. While students often complain about these ‘real-world’ examples that cannot be addressed simply by using a rote approach to physics, many more students have realized that conceptual struggles inspired by these questions have led to a deeper and more meaningful experience during the semester.
Buoyancy, aka ‘Archimedes’ Principle’ has always been a straightforward, but difficult, topic in particular. A simple way to express this is to say that the force of a fluid acting on an object either partially or wholly immersed in it is a) directed upwards, and b) equal to the weight of the fluid displaced by the object. Two seemingly innocuous questions from the ConcepTests section of Peer Instruction (pp. 170-171) in particular have frustrated my students:
Question 1 Question 2
Analysis of Question 1
In each case one must consider the change in the buoyancy force due to changes in the physical situation. For Question 1 the correct answer (1. Rises) can be qualitatively obtained by asking “What change occurs to the amount of fluid displaced?”. In the “before case”, the presence of the boulder in the boat requires that an amount of water equal to the weight of the boulder be displaced—i.e. , the boulder ‘floats’ by virtue of it being in the boat and the boat, in turn, being able to displace even more water due to its shape. In the “after case” the boulder, sinks to the bottom of the lake. This happens only if the boulder is denser than the water. Since the buoyancy force now acting on the submerged boulder is less than the weight of the boulder, it will accelerate downward, thus sink to the bottom of the lake. As result, the boulder is causing less water to be displaced than it did before while in the boat, so the water level in the lake must drop.
I sought a demonstration to use to illustrate this to unbelieving students. They had a persistent misconception that the buoyant force was constant independent of how the rock and the water interacted. In my demonstration, I cut off the top of a one or two liter plastic bottle, I place two or three metal pieces into a Styrofoam coffee cup, and then I place the cup into the bottle. I then fill the bottle with enough water until the cup floats free of the bottom of the bottle. I then record the water level by using a permanent marker: Figure 1. I then remove the cup and its contents, and transfer the masses one by one into the water—they sink immediately to the bottom of the bottle. Once the last mass has been deposited, I have the students note the much lower water level in the bottle. Then I place the Styrofoam cup in the water. Although the cup floats with hardly any additional displacement of water, students who expect the water to rise to the original level will frequently gasp with surprise as it remains at the lower level: Figure 2. I generally allow students to repeat this demonstration for themselves after class.
Figure 1: Floating cup with masses inside. The arrow indicates the current level of the water. Figure 2: Floating cup with masses on bottom. Note the large reduction in the water level!
Archimedes’ Principle states that the buoyancy force, FB, is equal to the weight of the fluid displaced. If the density of the fluid is rf, the acceleration of gravity is g, and the volume displaced is VD, then FB = rf g VD. When the copper is in the cup, it causes approximately 9 times its volume of water to be displaced as the density of copper is approximately 9 times that of water. After the copper has been removed from the cup and placed at the bottom of the bottle, it only displaces a volume of water equal to its volume. Thus there is a (9-1) Vo = 8 Vo overall decrease in the volume of displaced water.
Copper sinks to bottom Before:
Copper ‘floats’ while in the cup
After: Copper sinks to bottom
Before: Copper ‘floats’ while in the cup
Analysis of Question 2
I struggled to find a demonstration that worked well for Question 2. The correct answer (1. Moves up), is directly obtained by a thought experiment. When the ball floats on the water, it is supported by the buoyancy force of the water acting on the part of the ball below the surface of the water and by the relatively small buoyancy force of the air acting on the part of the ball above the surface of the water. The total of these two forces must equal the weight of the ball in order for it to remain in equilibrium. Imagine then that the ball is held in the position by a small wire finger while oil is poured into the container until the ball is completely immersed in oil throughout the experiment. If the small wire finger has kept the ball in its original position, what change has happened to the buoyancy forces on the ball? The force due to the water is unchanged since the ball has not been allowed to move and so the displaced volume of water is unchanged. However, the buoyancy force of the air on the part of the ball above the water has been replaced by the buoyancy force of the oil on the same part of the ball. Since oil is much denser than oil (has anyone ever seen salad dressing floating in air?), the buoyancy force must be greater. If the wire finger is removed, the ball will experience a net upward force, and thus accelerate upwards. Since the densities of the oil and water are nearly the same, the ball will have to move up a significant amount before equilibrium is restored.
The challenge that I faced was to find an object that would almost sink in water and yet almost float in oil. As a result, the specific gravity of the object should be between that of olive oil, about 0.09150-0.9180, but less than that of water, 1.000. I experimented with several sporting balls around the demonstration facility in the hopes of finding a relatively common object that students might have seen before (rather than an exotic and perhaps expensive polymer from McMaster-Carr). I found through trial and error that a field hockey ball was almost what I wanted. It barely floated in olive oil. To remedy this (and increase its overall density just a bit) I epoxied a large brass machine nut onto the label on the ball. This caused the ball to sink in the oil and yet barely float in water. With the densities of the three components thus nearly equivalent, this demonstration makes the physics principles involved much clearer for students.
I add a bit of food coloring to the water to enhance the contrast for students in the lecture hall as seen in Figure 3. Colored water is added until the ball clearly floats free from the bottom of the bottle. This provides a nice contrast to the naturally yellow olive oil after it has been added as seen in Figure 4. Olive oil is added until students can see that the ball is entirely submerged.
Figure 3: Field Hockey ball floating in mix of air/water. Figure 4: Field Hockey ball floating in mix of oil/water
A persistent misconception that remains after this demonstration is the idea that the ball should sink lower into the water due to the increased pressure from the oil poured above the ball. It is very difficult for me to convince students that Archimedes’ Principle does deal with either the issue of total pressure or the issue of pressure variation with depth. These concerns have no effect on the final result of Archimedes’ Principle for relatively incompressible fluids. I point out that the increased pressure of the layer of oil is also transmitted to the water beneath the ball—thus the pressure of the water that pushes up on the ball also increases due to the oil on top. This increase is in pressure is precisely the value required to provide the difference in buoyancy forces between the air and the oil on the ball with the ‘imaginary wire finger’ in place.
Calculating the final algebraic answer for this ‘compound fluid’ would be a bit tricky and detailed—I believe the qualitative analysis that I present in the previous paragraph is more direct. I sometimes illustrate the independence of this example from the pressure due to the variable height of the oil layer by adding more and more oil to the container. While the oil level rises in the bottle, the ball does not rise relative to the water!
Many thanks go to my students who teach me something about physics and life in each class, and to the members of Tap-L and PIRA communities who inspire me and provide wisdom when I cannot figure things out on my own.
One-Liter Plastic bottle, coffee cup, metal objects-- Found in recycling bins ~Free
Field Hockey Ball-- Found in street $6.99
Olive Oil-- About 1/3 liter $0.73
Nut-- ½-13 or any appropriate size $0.78
3 Minute Epoxy-- One pack $1.27
-- One can use “Glass Clear” Nalgene Beakers that are available from
http://vwrlabshop.com/graduated-griffin-beakers-pmp-nalgene/p/0006662/ Price Pk. 3 for $83.47
-- Rocks or other dense objects can be placed inside the cup. [I don’t recommend sand however, students may believe either that some has dissolved, or that air bubbles have been trapped by the sand (though this would reduce the observed effect!).]
-- Other cups can be used—I liked the contrast of the white/colored Styrofoam with the water.
Bonus Activity: Wacky Pennies—What Happened?
Sometimes I use buoyancy to introduce elementary school students to the idea of statistics and experimental control. I’ll have them place identical paper cups or other small flotation devices onto a pan of water, and then give them bags of pennies. I then challenge them to see how many pennies they can put in their ‘boats’ before the boats sink. I let them repeat this several times and record their measurements and observations. Hopefully they will learn about center of gravity, center of buoyancy, stability, and impulses delivered by dropping a coin versus placing it carefully in their craft. After several rounds each team will have determined their optimum number of pennies. I then have them call out their results and record them on the whiteboard. Surprisingly, although the data varies a little between groups, the data appears to separate into two distinct groups! Why the discrepant event? Because the pennies given to the students were pre-sorted into pre-1982 and newer pennies! The former are solid copper and have a mass of ~3.1 grams and the latter are copper-coated zinc disks with a mass of ~2.5 grams.
Recently an outreach group reported back after using some liquid nitrogen and dewars. One of their favorite activities was to break apart new pennies. Well, I tried this by cooling the new pennies, but nothing happened on re-warming. They then told me to hit them with a hammer: I then ‘whacked’ them using a 2 pound hammer with the cooled pennies on a lead brick. I’ve included the results in an attached plastic bag—all pennies were treated to the same cooling/whacking process. The photo below shows the apparatus and what happened…..
 In the event of motion in an accelerated reference frame, ‘upwards’ is directed opposite to the direction of effective gravity, where geff = g – a. “g” is the gravitational field in an inertial (non-accelerated) frame of reference, and “a” is the acceleration of the non-inertial frame of reference.
 Reduce, Reuse, Recycle
 Also found at Dick’s Sporting Goods http://www.dickssportinggoods.com/product/index.jsp?productId=971509
 On sale at Harris Teeter: 1 liter for $2.20
 McMaster-Carr, http://www.mcmaster.com/ ½-13” brass nut, PN 10 for $7.75
 McMaster- Carr PN 7538A11, Job-Size Epoxy Quick-Setting, 3.5 Gram Packet, Amber at $12.62 per Pack of 10