Pedagogue Padawan 180

In previous years, I’ve used old AP Physics B free-response questions for practice quizzes. With the shift to AP Physics 2, I’m still using these questions, but I’m adding an additional conceptual question to the end that requires students to justify their answer. Today’s practice quiz featured the classic example of a large tank of saltwater with a small drain with a plug and had students solve for the force on the plug, the velocity of the water once the plug breaks free, and the volume flow rate. What I added was:

The saltwater is replaced by an equal weight of freshwater. Indicate whether the speed of the water as it leaves the hole increases, decreases, or remains the same?

 Justify your answer.

Few students answered this correctly with the correct justification, and it lead to a great discussion. Some students realized and shared that the velocity of the fluid leaving the drain was only dependent on the height of the fluid in the tank and not the density of the fluid. (They basically derived Torricelli’s theorem.) Few students, however, appreciated that the saltwater was being replaced by an equal weight of freshwater. In order to have an equal weight of freshwater, a greater volume, and therefore greater height, of freshwater must be in the tank. Since the height increases, the velocity of the fluid leaving the drain would also increase.

It’s tough to anticipate the style of the questions on the new AP Physics 2 exam. I hope I’m on the right track.

In AP Physics 2, we whiteboarded the fluid dynamics problems today. This went better in that the questions and discussions were richer than when we whiteboarded the fluid statics problems last week. I expect this was due to two factors: one, students were better prepared; two, the problems were more challenging. In one class, we had a great discussion about why the pressure in an unconstrained stream must be atmospheric pressure. In another, we had a good discussion about how reasoning through a problem conceptually can provide bounds on possible answers that can be used to evaluate the reasonableness of the calculated answer.

  ##fluids ##whiteboarding  

AP Physics 2 students spent class exploring fluid dynamics through a series of activities. Using the Venturi meter is one of my favorite as it combines the model from fluid statics with Bernoulli’s equation for fluid dynamics. In addition, it demonstrates the effect of viscosity and one of the limitations of our ideal fluid model. One of the other activities is to blow a dime off the table and into a cup:

In the past, I’ve reinforced a simple (and incorrect) explanation through a misapplication of the Bernoulli Effect along the lines of “the dime lifts off the table because the Bernoulli Effect states that the fast moving air above the dime has lower pressure and the stationary air below the dime has high pressure.” My newly found deeper understanding began when I noticed a single phrase in the definition of the Bernoulli effect in Knight’s College Physics (emphasis mine): “The pressure is higher at a point along a streamline where the fluid is moving slower, lower where the fluid is moving faster.” The Bernoulli effect can only be applied along a streamline and not between streamlines. In addition, the Bernoulli effect focuses on a pressure gradient accelerating a fluid, not just the pressures associated with “fast” and “slow” fluid streams.

So today, I focused on how the streamlines are diverted over the top of the dime. The closer together the streamlines, the faster the speed of the fluid (justified with the equation of continuity – conservation of mass). The velocity of the fluid (air, in this case) increases as it is diverted over the leading edge of the dime and decreases over the trailing edge. The key factor is not that there is fast moving air over the dime but, rather, that the velocity of the air increases as it flows over the leading edge of the dime. Therefore, there must be a region of lower pressure over the dime.

Students were generally pleased with this explanation, but a couple were unsatisfied because we didn’t explain the cause of the region of low pressure over the dime. After reflection, discussion with my colleague, and some research, I think the best conceptual explanation focuses on the path of a particle along a diverted streamline. Since the path is curved, the particle experiences a radial acceleration toward the center of the curve. This radial acceleration requires an unbalanced force and the only force applied to this particle is a pressure difference perpendicular to the particle’s velocity. Therefore, on the concave side of the curve there must be a lower pressure than on the convex side of the curve. The combination of this effect across multiple streamlines results in the low pressure region over the dime.

Tomorrow, I’ll see what my students think about this perspective.

  ##fluids  

Honestly, clearly explaining Bernoulli’s equation has not been a strength in my instruction. I’ve done a better job each year making the connection to conservation of energy, and more and more students grasp the concept, but I have never done really well. Finally, I think I may have found a clear approach that demystifies Bernoulli’s equation through the use of LOL energy diagrams!

What set the stage for this insight was this excellent diagram in Knight’s College Physics text:

With this clearly defined system, I was able to focus on the volume of water on the left side of the system at time t1 “before” and the volume of water on the right side of the system at time t2 “after.” I treated the kinetic and potential energy of the middle region of the system as a different quantity (K + U) since it unchanged. I scanned my notes which includes the LOL diagram and the derivation of Bernoulli’s Equation from the LOL diagram:

Download (PDF, 943KB)

  ##fluids ##representations  

In AP Physics 2, students finished the statics portion of the fluids unit with the Archimedes Lab. In this lab, students apply their understanding of hydrostatics to experimentally measure the density of water by plotting buoyant force vs. depth of cylinder submerged. A lot of attention is placed on the accuracy of the measurements; we broke out the Vernier calipers to measure the cross-sectional area of the cylinders to be submerged. This lab will be the first test of the new lap report workflow with Chromebooks. Students will make their plots in Plotly, write their lab reports in Google Docs (embedding an exported image and link to the Plotly plots), and submit their report via Google Drive to Canvas for feedback and scoring.

  ##fluids ##practicumlab ##tech ##chromebooks ##plotly ##canvas  

Today we checked our understanding of static fluids with a quiz. I’m modifying AP Physics B free response questions to reflect what I expect the new AP Physics 2 free response questions will be. This usually involves fewer step-by-step prompts and a greater emphasis on justification of application of the concept. We again used InfuseLearning for peer instruction. I love that InfuseLearning supports ranking tasks, but am getting frustrated that I get disconnected at least one every class period which disrupts the flow. Time to file a support ticket.

  ##fluids ##peerinstruction  

Today was the first problem-focused whiteboarding session of the year today in AP Physics 2. I think it is a good sign that it is the eleventh day of school and it is the first day that we are plugging numbers into equations in class. I’m being very deliberate this year in the problems that I choose for us to whiteboard in class. I’m trying to select problems that integrate multiple concepts and lend themselves to clarification and extension questions. This was a good whiteboard for the first problem:

The problem using the Siri equation and Archimedes’ Principle to estimate a persons body fat percentage. I modeled a sample extension question by asking the group, “if the person was submerged in oil instead of water (ick!), would the weight underwater increase, decrease, or remain the same?” I then followed up with a challenge when the group responded that the buoyant force on the person would change by asking, “but yesterday we said that the buoyant force on the hydrometers were the same in all fluids. Can you remedy this apparent contradiction?” I’m trying to coach the students into using these whiteboard problems as prompts for conceptual questions that require careful reasoning and justification.

While students were whiteboarding, I tried the new Magic Marker app from Instructure to track three outcomes:

1. prepared before class
2. effectively presents whiteboard
3. asks insightful questions

I’m still undecided if gathering this information in this manner will be useful, which means I’ll at least try it again. What I can’t figure out is how to adjust the scores afterward since I accidentally swiped the wrong student a couple of times.

  ##fluids ##whiteboarding ##tech  

Today in AP Physics 2, students refined their conceptual understanding of buoyancy and Archimedes Principle through the use of a hydrometer to measure the specific gravity of three different liquids. In addition, this was our first opportunity to use Crocodoc to annotate a PDF that I distributed for the activity. One issue that we encountered is that we couldn’t figure out how to select a drawn annotation on the Chromebook so it could be deleted. Clicking on the annotation in Safari on my Mac worked fine. While students working on the Chromebooks at their desks works fine, it was a bit awkward to carry them from station to station as opposed to carrying a notebook.

This year I added a new question to the activity handout:

In which liquid does the hydrometer experience the greatest buoyant force? Justify your answer.

Of eleven groups, only two had the correct answer and only one could justify it with strong evidence. This question, and the resulting discussion, is what makes this activity so valuable. In one class, as a student justified his group’s incorrect answer, I could see the realization once he heard himself explain that the weight of the hydrometers is the same in all liquids. I love how we keep returning to the balanced force particle model as we work our way through fluid statics.

  ##fluids ##chromebooks  

Today in AP Physics 2, we derived the equation for the buoyant force on an object by applying a combination of the unbalanced force particle model (UBFPM) and our static fluid pressure model. We then discussed Archimedes Principle conceptually by focusing on balanced forces. We took a break from these derivations and discussions to examine Heron’s Fountain and try to figure out how it worked.

After some careful observations, the class, usually led by a couple of students, figured out the mechanism. The fountain also provided an opportunity to discuss conservation of energy (is the fountain a perpetual motion machine?) which also was raised in the discussion of Pascal’s Principle applied to hydraulic lifts.

AP Physics 2 started class sharing their mathematical models for pressure vs. depth below the surface of a fluid. In general, students struggled a surprising amount getting the equation for the best-fit line correct and handling unit conversions. One student shared that they needed a week to get back in the swing of things (it is the eighth day of class), and another shared that them struggling with this was “kind of sad.” Once we got the mathematical models captured in the Google Doc correctly, I was surprised at the range of values for the slope.

We discussed potential sources of major systematic error such as losing the seal between the glass tube and the plastic tube or heating up the air trapped in the plastic tube by holding it tightly throughout the experiment. I need to investigate calibrating the pressure sensors and seeing if the experimental technique can be improved to gather higher quality data.

After discussion the lab, we derived the general form of the equation through application of the balanced force particle model on a cylinder of water at rest within a beaker of water. It was good to reinforce a basic model from last year.

  ##paradigmlab ##setbacks ##expdesign ##fluids