Taking the hydrostatic paradox to the next (water) level

How well do people understand hydrostatics? I am preparing a workshop for tomorrow night and I am getting very bored by the questions that I have been using to introduce clickers for quite a lot of workshops now. So I decided to use the hydrostatic paradox this time around.

The first question is the standard one: If you have a U-tube and water level is given on one side, then what is the water level like on the other side? We all know the typical student answer (that typically 25% of the students are convinced of!): On the wider side the water level has to be lower since a larger volume of water is heavier than the smaller volume on the other side.

Clearly, this is not the case:

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However, what happens if you use that fat separator jug the way it was intended to be used and fill it with two layers of different density (which is really what it is intended for: to separate fat from gravy! Your classical 2-layer system)?

Turns out that now the two water levels in the main body of the jug and in the spout are not the same any more: Since we filled the dense water in through the spout, the spout is filled with dense water, as is the bottom part of the jug. Only the upper part of the jug now contains fresh water.

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The difference in height is only maybe a millimetre, but it is there, and it is clearly visible:

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Water level 1 (red line) is the “main” water level, water level 2 (green line) is the water level in the spout and clearly different from 1, and water level 3 is the density interface.

We’ll see how well they’ll do tomorrow when I only give them levels 1 and 3, and ask them to put level 2 in. Obviously we are taking the hydrostatic paradox to the next (water) level here! :-)

Pressure distribution on a wing

Another neat experiment in the collection I’ve recently been talking about is measuring pressure at different points on a wing profile. It’s not terribly surprising that – as long as the wing is oriented in the correct way in the flow – pressure is high below the wing and low above it. Kinda the whole point of having a wing profile. Yet, it’s nice to actually measure it.

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Measuring pressure at different points on a wing profile

And yes – next time I set up that manometer I’m gonna make sure that it’s a little easier to get a good reading!

Playing with Venturi tubes II

On Monday I posted about playing with Venturi tubes. Guess what: We are going to play more today! Because today the Venturi tubes are connected to a “proper” manometer:

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Venturi tube connected to manometer

Now, if I wasn’t so lazy this would be a great opportunity to get good readings of the pressure differences caused by different flow rates. However, I’ll just let the images speak for themselves. Enjoy!

 

Playing with Venturi tubes

A Venturi tube is one of the things one hears about in hydrodynamics class all the time, but one never gets to see them for real. And even though I just said on Friday that the thing that I found most fascinating in the aerodynamics collection I got to borrow recently was to see how the flow reversed downstream of a paddle I might have to take that back, because the hands-down most exciting thing was to play with a Venturi tube!

So what is all the fuss about? This is what a Venturi tube looks like:

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Venturi tube in the no-flow state

Basically, it is a tube, open at both ends, that gets thinner in the middle and wider again. All the rest you see in the picture is props: The mouth of the fan in the top right, and then three U-tubes filled with dyed water below the Venturi tube.

The Venturi tube is so famous because it nicely demonstrates the Venturi effect, namely the reduction in pressure that occurs when a flow is accelerated. In the case of the Venturi tube, the flow is accelerated in the thin section of the tube, where – for continuity reasons – it has to go faster than in the wider sections. So what happens when we turn on the fan?

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Venturi tube. Pressure decreases in the thin section of the tube, visible by the red water being “sucked up” in the middle U-tube

Yep! The levels in the three U-tubes change. And most importantly, the pressure for the middle U-tube drops, as demonstrated by the red water being “sucked up” on the side of the U-tube that is connected to the Venturi tube.

Watch the movie below to see this in action:

Letter tubes and hydrostatic pressure

How less than 25% of the tested students give consistent answers to these problems.

This is already the third blog post talking about the paper “Identifying and addressing student difficulties with hydrostatic pressure” by Loverude, Heron and Kautz (the first two posts here and here). But I am still a bit in shock by what I read in that paper.

Consider the figure below. A N-shaped tube filled with water.

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The N-tube problem.

Students are asked to rank the pressure at points G, X, Y, Z.

Because I hate reading electronics papers where they give you the questions and the students’ misconceptions, but don’t tell you what the correct answer would be (how would I know?) I am going to give you the answer, but I’ll assume that you know it anyway. Clearly, points X, Y and Z have the same pressure, whereas the pressure at point G is less.

So what do students say?

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The N-tube problem and the typical WRONG student answer.

A very prominent answer, according to the authors of the study, is that students confuse pressure with weight. Since there is more water above X than above any of the other points, the pressure here seems to have to be highest. And following this logic, the pressure at Z is the smallest (for a sketch of the wrong “h”s that go into this answer, see the figure above).

Using a different-shaped tube, and asked again to rank pressures, students find different results (rather than giving the correct answer, which would be that the pressure at X and Y is the same, the one at W is higher and the one at Z is lower):

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The U-tube problem.

Here, many students conclude that the pressure must be increasing from X through W through Y through Z, hence perceiving pressure as varying along the curvature of the letter.

When students in that study were shown both letters together, this is what the typical answers look like:

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Comparing the N-tube and the U-tube.

The authors find that less than 25% of the students answer these two problems (even when shown side-by-side) consistently. And consistently means just that: They either answer both correctly, OR they answer both of them based on the misconception described for the N-tube, OR they answer both of them based on the misconception described for the U-tube.

This means that 75% of the students in the study didn’t even have a mental model that they consistently used. And those were students who had gone through the standard instruction in hydrostatics. This makes me wonder how this translates to my own students. I have never explicitly talked about these kinds of problems, assuming that students had a full grasp of the material. But clearly this is an assumption that should not be made. But where do we have to start teaching if this is still so fraught with difficulties? Do you have any ideas? Then please let me know.

Barometer problem.

Still talking about hydrostatic pressure.

Yes, I am not done with hydrostatic pressure yet!

One of the problems students were given in the study “Identifying and addressing student difficulties with hydrostatic pressure” by Loverude, Heron and Kautz is a barometer problem.

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Barometer problem – compare the pressure at point x and y.

Students are asked to compare the pressure at point X and point Y. Apparently, this is not as obvious as it seems to me. So before I go into the detailed discussion (I might do it in a later post – anyone interested in reading it?), I thought I’d just set this up. Because to me it seems that if you see this sitting there with the liquid clearly not moving one way or another, the solution has to be clear. We’ll see what others think, but here we go:

If you want proof that the tubes are open at the bottom and that there still is a hydrostatic equilibrium, watch the movie below. Spoiler alert: You might have fallen asleep by the time things start moving in the movie ;-)

Hydrostatic pressure

What are students not understanding about hydrostatic pressure?

Tomorrow (today by the time this post will go online, I guess) I will present the paper “Identifying and addressing student difficulties with hydrostatic pressure” by Loverude, Heron and Kautz at the Journal Club at work. So tonight I am trying out a couple of experiments that I would like to show with it.

I already know that I am not supposed to show the experiments during the talk, but I figure that there is no harm in having them prepared in case anyone wants to see them afterwards.

And good thing I tried them before instead of just assuming that they would work!

For the first experiment, I had this awesome idea to re-create something I saw as a child when on vacation on a farm:

I was clearly very impressed with it – this picture is from 1994 and I remembered it and asked my parents to track it down for me!

Anyway. Since I wasn’t sure if my colleagues would be happy with that amount of water on the floor, I decided to go for a smaller version of the same thing.

This is what I wanted it to look like (and what it looks like in my presentation):

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Hydro(almost)static pressure in the idealized case.

And this is what the experiment ended up looking like:

How disappointing! I guess the holes that I poked into the bottle aren’t well made. But good thing I tried. Watch the movie if you want to pay attention to if you ever want to present this experiment.

Yes. You want to use tape that keeps the water inside the bottle. Until you want to take the tape off. Then you wish you had used something that actually comes off…… ;-)

Cartesian divers – theoretical considerations

 A bit more reflection on cartesian divers.

When I wrote the two previous posts, I had known cartesian divers for a very long time in many contexts, for example as something that is routinely used in primary school teaching. While I was aware that developing a correct physical description of such a diver is challenging, I assumed that everybody had an intuitive understanding of how a diver would react when pressure was applied on the bottle. To me, this is an experiment that I would use to demonstrate the different compressibilities of air and water, assuming that everybody can imagine what happens if the density of a floating body changes.
Turns out my assumption of what people intuitively understand was way off. In the paper Helping students develop an understanding of Archimedes’ principle. I. Research on student understanding”, Loverude, Kautz and Heron talk about difficulties university science majors have with hydrostatics. Of seven volunteers who were interviewed, who had all completed their instruction in hydrostatics and all reported course grades at or above the mean, all but two predicted that the diver would rise as pressure was applied to the bottle. And none of the students could account for the observation that the diver sank!
Now I’m wondering at which point the students’ difficulties arise. Is it that they don’t know about different compressibilities or is it at a much more basic level? From the study mentioned above it seems that students don’t appreciate the tiny density range (where calling it a range might already be over-stating it) in which a body can float in (non-stratified) water without swimming at the surface or sinking to the ground. In a way this makes sense – most of the time that we look at water in a way comparable to how we look at a cartesian diver (i.e. through side walls so we are looking at a depth section of a non-stratified fluid), we are actually looking at aquaria where fish float in very similar ways to the cartesian divers. But we never stop to think about how floating and adjusting depth in a fluid is actually quite an achievement. Which we see when the fish die and first float at the surface and then sink to the bottom…
In any case. If it is the case that students don’t appreciate how rare it is for something to float in a fluid, then showing a cartesian diver might even be working against us by reinforcing a perception that is harmful to the students’ future understanding of hydrostatics. Or we can use the divers in a different way – have students build them themselves, so that they need to fiddle with them to adjust their initial density until it is just right, before they start working in the way shown in the previous posts. I think this is a thought I want to develop further… So stay tuned!

Cartesian diver – organic version

Using orange peel as cartesian divers.

Guess what my mom told me when we were playing with cartesian divers the other day? That orange peel works really well as a cartesian diver! Who would have thought?

And just because we like playing we tried both orange peel and tangerine peel. Watch!

Funnily enough, they behave very differently. While the thick orange peel works really well, the much less thick tangerine peel very quickly looses all the air bubbles and hence the buoyancy and the ability to adjust buoyancy. So if in doubt (and not interested in extending the experiment to a lesson in contrast and compare) – oranges are the way to go!