Category Archives: AP2

AP Physics 2 Syllabus

I was recently able to get my syllabus approved by College Board. The approval number is # 1485589v1 Authorized

You can download a copy of the .docx version here.

Thermodynamic processes

I’d like to show how the “Super LOL Diagram” can be used to help solve problems for thermodynamic processes. As an example, imagine the problem tells us that a 1 cubic meter sample of gas at a pressure of 101 kPa, is compressed isothermally at a temperature of 300 kelvin to half its original volume.

Sample super lol2

The first thing we can do is write the given information, which is done in red ink. Next we can try to figure out the rest of the variables for the initial and final states. From the Ideal Gas Equation

PV = nRT

we can calculate the number of moles at the initial state (blue ink). If we assume it’s a sealed container, then the moles would be the same in the final state (pink number). We were told the process is isothermal, so the final temperature must be the same as the initial temp (orange ink). We can determine the pressure at the second state by again using the Ideal Gas Equation.

The next step is to draw the approximate size of each container below the state variables. Since the initial volume is twice that of the final, we make the container on the left about two times bigger than the one on the right (grey ink).

Next up is the energy present at each state. To find these values, we can use the other equation developed during the computer simulation/paradigm lab:

U = \frac{3}{2}nRT

Since this only depends on the number of moles (a constant) and temperature (not constant) the relative size of the energy bars will be proportional only the temperatures  of the two states (purple ink).

The next step is drawing the PV diagram (brown ink). This shouldn’t be anything new for people familiar with thermodynamics or from most textbooks. For the sake of brevity, I’m not going to explain that. After we make the PV graph we can now determine the working that occurred between the system and the surrounding by finding the area under the curve (brown shading). To find the area, we can use calculus or the equation given in most textbooks for the work of an isothermal compression:

W = -nRT ln \frac{V_1}{V_2}

The last step is to complete the energy flow diagram (the “O” of the “LOL Diagram” at the bottom). By having the grey pistons drawn, we have a clear indication that the gas was compressed, which would mean that energy flowed into the system from an outside force moving the piston inwards (black ink). To determine the amount of heating, we can use the First Law of Thermodynamics equation:

\Delta U = Q + W

We know the initial and final energy (purple ink), we know the work done (black ink), so we just have to calculate the heating by subtraction (dark green ink).

System of Ideal Particles Model

Model Development

To begin this unit, we perform three experiments that together make up the paradigm lab. The first is connecting a syringe to a pressure sensor. Students record the volume from the scale on the syringe and note the pressure at that volume. As they then compress the plunger, they record the new volume and pressure.

The second experiment is to see how temperature effects pressure. The teacher notes show using a metal container. I couldn’t find one, so I used a glass bottle. Put a two hole stopper into the opening. In one opening, insert a digital thermometer, in the other, a valve to connect the pressure sensor. (Note: if you can’t get the thermometer to make a tight seal, you can put it in the water bath.) Now take the bottle and (partially) submerge into boiling water. I used a clip attached to a ring stand to hold the bottle just above the bottom of the beaker. I had the beaker of boiling water on top of a cold hotplate with a magnetic stirrer at the bottom of the beaker. Record the temperature and pressure, and then add ice. Wait until the temperature and pressure reach equilibrium, and repeat.

I was originally going to start with ice water and heat it up on the ice plate, but was told over the summer by a teacher at my AP workshop that the stopper will likely pop off due to the increase in pressure. That is why you start from boiling and cool down.

The third experiment is a little trickier. Attach a two hole stopper to a rigid container (again, I used the glass bottle). In one opening, the valve to the pressure sensor, in the other to a three-way valve that can be connected to the syringe and open to the atmosphere. Fill the syringe to a certain volume (I used 20 mL). Open the valve between the syringe and the container (closed to the air) and inject the air into the container. Close the valve to the container, such that is open to the atmosphere, and fill the syringe again to 20 mL. Repeat several times.

The end result of these three experiments should show graphs similar to what was seen in the computer modeling experiment:

Pressure \propto \frac{1}{Volume}

The graph of pressure versus temperature is linear, with a positive intercept. This can lead to a discussion as to why the computer model was proportional, but the experiment was linear. The teacher should lead the students to the ideal of absolute zero. If you extrapolate the line, you can find the horizontal intercept which should be near ~ -273 ^oC. If you translate the temperature into units of kelvin, you should then see that:

Pressure \propto Temperature

and since the amount of gas added each time was constant, you can lead students to the conclusion that you were adding a constant number of particles each trial. The data should show a linear relationship. Once again, the teacher can lead the students to the conclusion that the vertical intercept represents the number of particles in the container at the beginning. If you were able to start with an empty container, then:

Pressure \propto Number

From there, the teacher can lead the students to the combined proportionality seen in the computer simulation:

P \propto \frac {N T}{V}

Although the computer simulation took about two weeks, I think it is worth it, as it gives the students a concrete visual model of what is happening. Most of my students have already taken both Chemistry and AP Chemistry. Although they already knew the equation for the Ideal Gas Law, most commented that it finally makes sense after working through this entire progression of the computer simulation and experiments.

 

Model Deployment

 

Worksheet 1

The first worksheet has the students practice using the two equations developed in the simulation and experiments:

P = \frac{kNT}{V} = \frac {nRT}{V}

and

U = \frac {3}{2} Nk_{B}T = \frac {3}{2} nRT

It also introduces isothermal, isobaric, and isochoric processes. To analyze these, the worksheet introduces students to what I called Super LOL Diagrams which look something like this:

Super lol
The bottom row is the LOL you know and love from the Energy Transfer Model. (If you don’t know and love it, let me introduce you to Kelly O’Shea and her awesome explanation of LOL diagrams).

The only flavor of energy present in our system is the thermal energy we developed in the computer model:

U = E_{therm} = \frac {3}{2} k_{B}NT = \frac {3}{2} nRT

The top row of the Super LOL is new. I hope to write post showing how to use this in greater detail (edited: see explanation of Super LOL here), but the short version is this:

Students should write all the values of the state variables (n or N, P, V, and T) where provided. They can use the Ideal Gas Equation if not all are given in the problem. The “U” shaped object below the state variables is a blank piston. Students shade in a qualitative picture of the piston before and after whatever process is occurring.

The “L” in between is for a P-V graph. With this, students can determine the amount of working done by finding the area underneath the path from state 1 to state 2. The pistons help reinforce the direction of working, if compressed the surrounding is working on the system. If the piston is expanding, the system is working the surrounding.

In the “O” of the LOL diagram, the system is the gas inside the piston. In addition to working, the teacher introduces heating as a second energy exchange that can occur between the system and the surrounding.

With this, we have a visual tool to represent the First Law of Thermodynamics:

\Delta U = Q + W

I’m holding off adding radiating as a third energy exchange until we get to light later in the course. I also am not going into conduction vs convection as I’m not sure how much this will come up. The AP equation sheet has the equation for conduction, but I guess we’ll see over the next few years how much it is used.

Worksheet 2

The second worksheet two builds upon the thermal processes in worksheet 1 by analyzing thermodynamic cycles. The Super LOL expands to become a Super LOL Box (I hesitate to call it the Super-Duper LOL, but maybe that’s a better name).

Super lol box
The center “L” is the P-V graph, the four corners are for each of the 4 stages in the cycle (You can make a LOL triangle for a 3 stage cycle or just cross out one corner of the box). The “O” in between is for the transition between those two adjacent steps of the cycle.

From here, the 2nd worksheet has students add up the total working in all the steps (W), the total heating added in all the steps (Q_H), and the total heating removed in all the steps (Q_C into a diagram of the overall energy flow:

Overall energy flow

Worksheet 3

Before attempting worksheet three, students are provided a reading that introduces 5 classic definitions of Entropy with proofs that they are all equivalent. It also introduces several of the common thermodynamic cycles commonly called Carnot Cycles and Otto Cycles. It also explains the equation for the change in Entropy:

\Delta S = \frac{Q_H}{T_c} - \frac{Q_H}{T_h}

One of the things I like about the teacher notes is that they stress explaining Entropy as the spreading out of energy into the different degrees of freedom of the system.

With my students we discussed the $\frac  {3}{2}$ in the thermal energy equation that more complex particles (aka molecules) will have a greater fraction (e.g.: $latex \frac {5}{2} for diatomic molecules). I did that at the time of the lab this year, but may hold off until this point of the model deployment in the future.

By this logic, Entropy is similar to diffusion studied in chemistry (at least at our school it is) and heating in that the quantity flows from high concentration to low concentration. I think this makes Entropy easier to understand than the classic concept of Entropy as the “disorder” of the system.

In the worksheet, the students are led to see the similarities to heating in that we can try to predict what will happen as Entropy can only flow from high concentration to low, never the other way. If you need it to flow the opposite way, you must have a second process that provides work, and in the process has (an equal or) a greater flow of Entropy, what is commonly called the Second Law of Thermodynamics.

vPython Modeling

To open the AP2 course, the students will learn how to make a computer simulation using vPython over about 2 weeks of classes. I would like to thank John Burk’s awesome series of blogs. I’m also pulling from resources found in the additional materials from the AMTA website.

The first part of the unit is learning some of the basics of computer programming, how to create objects, change attributes, etc. The fun begins when they learn how to make an object move in one dimension. They begin to see how the physics we learned in the AP 1 course is used in the code. After learning how to make a sphere move, they quickly proceed on to make that ball bounce with a perfectly elastic collision off a wall. From their, they add a wall on the opposite side to make the ball bounce back and forth. Then they learn how to adjust the code so that the ball bounces inside a box with motion in all three dimensions.

The next big step in the sequence is to add a second ball. To start off, we go back to 1D motion. They learn how to have the balls bounce with a perfectly elastic collision using a transformation into the Center of Momentum frame of reference to make the calculation easier. After that hurdle, we then move on to two balls moving in three dimensions. Once they can successfully do that, I ask the students what they would need to do to add a third ball. By this point (about a week into programming), they can already realize that they would need to copy the code for the ball interaction twice (ball 1 to ball 2, ball 1 to ball 3, and ball 2to ball 3).

Before they start doing that, I introduce lists to help clean up the code. They create a list for walls, and a second list of balls. Once I help them figure out how to write the math for the interactions between lists and among members of the list of balls, we can now start adding a lot of balls quickly.

The last step is to start making the code able to do the unit’s experiment. They set up parameters at the beginning of the code to input the size of the box, the average velocity of the balls, and the exact number of balls. For each one, I introduce the ability to create random numbers. For velocity, I also introduce spherical coordinates. They create a random velocity based on a set mean and standard deviation (random gaussian number), with random angles, then transform those into the x, y, and z components of the velocity for the code. I also show them how to have the code output the average kinetic energy of the balls, the total kinetic energy, and the pressure the balls exert as they bounce off the walls.

At first the lab groups are to vary 1 input parameter and record the other inputs parameters and the output results. As you begin running the program, and manipulating the given inputs, you can generate graphs from them.

Pressure vs number of particles

 

Pressure vs volume

We then define temperature as the average kinetic energy of the particles to create a Pressure vs. Temperature graph:

Pressure vs temperature

 

From the three graphs we can see that pressure is proportional to number of balls and Temperature, and inversely proportional to volume. If we then graph pressure vs. NT/V we get:

Pressure vs nt v1

We can then conclude create an equation:

P = k\frac{NT}{V}

I haven’t figured out how I need to tweak to code such that the slope in Boltzmann’s constant, k_b, but I’m guessing is has something to do with choosing the correct parameters. You can then discuss that the SI units for counting particles is the mole. This changes the above equation to:

P = \frac{k}{N_A} \frac{nT}{V}

Which can be simplified as:

P = \frac {nRT}{V}

You can also look at which variables effect the total energy of the system. The graphs that show something are:

Total energy vs number of particles

 

 

Total energy vs temperature

If we then plot Total Energy vs NT you get the following:

Total energy vs nt

If you then discuss the units of each of the combined graphs (P vs. \frac{NT}{V} and E_{total} vs. NT) you will see that the units of slope are the same. If you then make a new graph of E_{total} vs. kNT, where k is the slope from the P vs. \frac {NT}{V} graph, you get the following:

Total energy vs knt

From this graph we can see that the slope is very close to \frac {3}{2}, so we can create the equation:

E_{total} = \frac {3}{2} kNT

Since the only flavor of energy present in our system we can say that the total energy of the particles E_{total} is the  of the system U. Thus:

U = \frac{3}{2}kNT

Since I had to do this in a computer lab, I did all of this before we moved into the lab to begin the second unit, the System of Ideal Particles Model. If you have computers in the class room, you might want to delay discussing the combined graphs and subsequent equations until after completing the paradigm experiments.

AP Physics 2 Storyline

As I prepare for the transition to the AP 2 course, here is the story-line of how  I’m planning to teach my AP 2 students next year. Before I get into the details, I do plan to use Modeling Instruction throughout the course. If you haven’t had the chance to take a workshop, do yourself a favor and find one. Also, I plan to make future posts providing more detail for each unit.

For those that have drilled deep into the materials provided by the Modeling Community, you may have found the resources provided for topics such as fluids and ideal gas law. Within those materials, there is a recommendation to use computer programming to bridge the gap between systems with only a few particles (read: AP1 events) to systems with many, many particles (read: fluids and/or thermodynamics). With that in mind, I plan to begin the year with a unit on Computer Modeling. I hope to accomplish two things: 1) Review some of the major concepts from AP1, and AP2) Prepare the students so they can see how analysis of fluids, the Ideal Gas Law/Kinetic Molecular Theory all come out of the models we build in AP1.

From there, we move to a unit on Ideal Gases, which I call the Ideal System of Particles Model (ISPM). Within this model we will recreate some of the various classic gas experiments (Boyle’s, Charles’, etc.) to build a model for monatonic ideal gases. We will connect what we see in the lab, with what our computer models predict. We will also begin to look at what happens to that gas when changes occur such as a compression or an expansion. In so doing, we will enhance our model so that it can predict what happens to the energy within our gas with those various changes (read: thermodynamics).

After building a model for gases, we move on to a unit on fluids which I’ll call the System Flow Model (SFM). Again, I plan to use the modeling materials provided. For those that haven’t seen them, they in essence begin by looking at what is happen to a small volume of water. In so doing, we build the continuity equation and the energy density equation (Bernoulli’s Equation). We also look at the pressure at different depths, and how those with effect that small volume (buoyancy).

In unit four, we begin the a second major concept of the year, electrical interactions. We again will make use of the 4 units developed by the modeling community. First up for this concept, a model for an Electrically Charged Particle, which I will call the Charged Particle Interaction Model (CPIM). We build parallels to the other non-contact force studied in AP1, namely gravity. We develop a field equation and a universal equation (Coulomb’s Law), just as we did with gravity.

In unit five, we build upon that model to describe the energy changes that can occur, what I call the Electrical Energy Transfer Model (EETM). After quickly developing a model for the energy storage in electrical fields, we create a short-hand way of tracking energy changes by looking only at the product of the field with the distance through which it moved (electric potential).

In unit six, we now develop a model for the movement of charged particles through circuits, what I call the Charge Flow Model (CFM).  We begin by tying this model back to the third unit on flowing particles. Along the way, we also develop Ohm’s Law, and Kirchhoff’s Laws. We also  develop a means to predict the power dissipated by a resistor.

In unit seven, we begin by looking at a weird side effect of moving charges, namely, their interaction with other moving charged particles, the Magnetic Interaction Particle Model (MIPM). We show how this force a different interaction than the electric force, but also show how they are both based on the same fundamental quantity of charge. Along the way we build up a third type of force field, and look at the quantities that effect it and interact with it.

In unit eight, we begin our third major topic of the year, the study of light. In this unit we build a Particle Model of Light (PLM) to describe reflection of light. We look at both smooth and rough surfaces. Also study flat and curved surfaces.

In unit nine, we see the limitation of a particle model of light in understanding how lenses, diffraction gratings, and thin films. In the process we develop a new Wave Model of Light (WML). Along the way we develop a set of equations: one that relates the focal distance, image distance, and object distance; and a second that related the heights of the object and image with the distances of the object and image. We also develop ways of understanding total internal reflection, double slit or diffraction gratings, and the effects of thin films.

In unit 10, we encounter some events in which the wave model breaks down: photoelectric effect, atomic emission/adsorption of light. In the process we build a new hybrid “Photon” or “Quantum” Model (QM). By no means are we building the actual Quantum Mechanical Model through things like the Schrodinger’s Equation, but we are building the concept of photons and discrete energy levels within an atom which are further along than just the Bohr’s Model.

In the final unit, we will again do a mild “hand waving” to try to take our “Quantum” model and use it to explain radioactive events. Along the way we will develop our “Standard” Model (SM). Again, not the actual Standard Model developed of the last 5o years, but a rudimentary look into particle physics to study nuclear decays, Compton Scattering, and a very cursory look at the Strong and Weak Nuclear Forces.

As you can see, if you’ve made it this far, this course is not as completely developed at this point. I’ve taught some aspects of this. I plan to make use of as much of the advanced modeling materials as I can, but I’m guessing I’ll be creating some of the materials as I go. To end the year, I plan to have students do a second Video Project in which they must try to analyze videos using models from these second year  models to begin reviewing and getting ready for the AP2 exam.