Unit 4: Balanced Forces Particle Model

The fourth unit introduces Forces. It takes about two and one-half weeks. It begins with a series of Bowling Ball Activities; e.g. make the ball speed up, slow down, turn 90^{\circ}, move in a circular arc. In the previous unit, a simplified “Interaction Diagram,” what many call a “System Schema.” As we begin to use this diagram, we see that multiple interactions are occurring.

Students are asked to make a quick sketch of momentum vs. time for the bowling ball starting from rest, being pushed by the broom, and then rolling at “constant” velocity. Most agree that it should be a horizontal line at p = 0, then a diagonal line as it is speeding up, and then a horizontal line at the final constant velocity.

In previous units, students saw the slopes are often important quantities, so they are asked to try to figure out what this slope of momentum vs. time represents. Usually, with little guidance, they can figure out that this slope represents the rate at which momentum is swapping, which we define as a “Force.” Thus, if an interaction is the swapping of momentum, it’s derivative is a Force.

From there we introduce the different types of interactions present during our activities. We identify three contact forces: push (Normal), pull (Tension), and slide (Friction). We also name one non-contact force, Gravity. From there, these types of interactions are added to the interaction diagram. Then show how this interaction diagram, can be used to make Force Diagrams.

We conclude the day by noticing that multiple interactions can balance out, thus multiple forces on a single particle or systems can balance out. We go on to notice that the system will only accelerate when the forces are unbalanced, which I mention is Newton’s 1st Law (when in an inertial frame of reference).

So the next step in the sequence is Worksheet 1 in which they practice making the Interaction Diagram with forces included and Force Diagrams. They are aided by a reading that help show in greater detail how to create Force Diagrams including equality marks.

From there, we do a lab to try to begin to understand the non-contact force of gravity by hanging various masses on a spring scale. Thus finding the relationship between mass and the force of gravity.

Once that’s complete we add this calculation to begin to add numerical values into Force Diagrams in worksheet 2. Along with a second reading, we end the unit with worksheet 3 as we add component forces into the mix.

The student goals for this unit are:

  1. can draw a properly labeled free body diagram of all the forces acting on an object including equality marks
  2. given one interaction between two objects, determine the direction of force exerted on each object
  3. determine the direction of acceleration of an object from a free body diagram
  4. determine whether or not the forces are balanced given information about the motion of the object
  5. determine the force of gravity on an object within a gravitational field

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.

AP Physics 1 Syllabus

I recently was able to get my syllabus approved by College Board. The approval number is # 1485588v1 Authorized

You can download the .docx copy 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.

Unit 3: Momentum Transfer Unit

This third unit on the MTM is the first significant deviation from the traditional modeling framework. I expect it to take approximately two weeks. It will begin with two carts “exploding” apart and whiteboard meetings to analyze the results (ratio of masses of carts -> ratio of velocities of carts).

From there they will proceed through some of the modeling materials for the Momentum Transfer Model as provided by the Modeling Materials. The main focus of the unit will be that momentum is a quantity that is swapped between objects, depicting those swaps with “interaction diagrams” (formally called system schema) (labels of types of forces withheld during this unit), and momentum diagrams (IF charts). Also along the way, I will try to emphasize the similarity between displacement (being term for a change in position) and impulse (being the term for change in momentum).

The first worksheet is the same as the first worksheet from the Modeling Materials. It looks at mainly qualitative events and has the students determine relative momenta or impulses. We then look numerous collisions to see if momentum is conserved in collisions as well as the explosions seen in the lab.  We skip the second worksheet provided by the Modeling Materials, as most of these problems focus on calculating impulses from Force & time. These types of problems will be address later in the UFPM unit.   The new second and third worksheets use momentum diagrams to solve collision problems. We end with additional problems for review.

The students goals for this unit are:

SWBAT

  1. create an interaction diagram including the identification of the system.
  2. create a momentum diagram (IF diagram) for an event.
  3. interpret a momentum diagram by creating a mathematical model of an event.
  4. correctly solve problems involving an exchange of momentum.