# Category Archives: Momentum

## AP1 Unit 8: Rotating Bodies Model

We begin this unit with a simple paradigm activity similar to Unit 4‘s bowling ball games. We begin with a dot on a pulley. I begin rotating the pulley and tell the students to imagine the disk is spinning at constant speed. I ask them to figure out a way to describe the motion of the dot as a function of time. I give them a few minutes to talk as a group. Since my students math experience ranges from Algebra II to Calculus, I will usually have several students who come up with:

$x = r cos \left( \theta\right)$

$y = r sin \left(\theta\right)$

To ensure all the students remember the basic equation, we then discuss the parent function for sine and cosine.

$y = Acos \left(Bx + C\right) + D$

To do this, I first pull up Desmos and show the effects of changing each of the four variables. Here’s what happens when you increase $A$:

Here’s what happens when you increase $B$:

Here’s an increase in $C$:

And an increase in $D$:

We discuss the meaning of $A$ as the amplitude or vertical stretch of the graph, $B$ as the “frequency” or horizontal stretch of the graph, $C$ as the horizontal shift of the graph, and $D$ as the vertical shift. Since we have discussed frequency last unit, I relate that $B$ is not exactly frequency, but that it is related to frequency, and tell them that we will go into more detail later this unit.

From there, I grab a string with a mass tied at one end, I grab the other end of that string and attach it to the edge of the pulley. I then wind the string around the pulley until the mass is touching the side of the pulley. I then challenge them to come up with a way to describe the dot as a function of time as I let go of the pulley and the mass falls to the table. I give them a few minutes to struggle with coming up with a function. After of few minutes of struggle, I suggest looking at the world from a different perspective. I then ask them what other ways could we describe this beyond using $x$ and $y$?

I socratically lead them to using polar coordinates. Since the distance of the dot to the center of the pulley doesn’t change, the motion can be described as a one dimensional motion using only $\theta$. I attach a photogate to the stand, rewind the string, and display the results of $\theta$ vs. time. We usually get a graph that looks like this for $\theta$ vs. time:

<I always have trouble getting the timing of dropping the mass and starting the data collection, so my graphs are usually shifted like this one>

As we discuss the graph, the students usually notice that the graph looks similar to what we saw in Unit 2 other than the time shifted (in this example by a little less than 0.4 s). So I ask them again to mathematically describe this graph. They will usually then say that it should look like:

$x = \frac{1}{2}a_o t^2 + v_o t+ x_o$

When asked what the slope and curvature represent, to which they responded it should represent the “velocity” and “acceleration.” At that point I tell them that we call it “angular velocity” and give it the symbol: $\omega$ and “angular acceleration” which gets: $\alpha$. I then show them the graph of $\omega$ vs. time:

With this, they can see that we get a fairly linear relationship where the slope is the angular acceleration:

$\omega = \alpha_o t + \omega_o$

We then proceed through the same derivations to get to find the area under the curve.

$\Delta \theta = \frac{1}{2} \left(\omega_o + \omega\right) t$

Then use that to find the equation for the $\theta$ vs. time graph.

$\theta = \frac{1}{2}\alpha_o t^2 + \omega_o t + \theta_o$

After this day of work, we progress to worksheet 1, which deals with constant angular velocity problems, then worksheet 2, which deals with constant acceleration problems.

From there, we move on to the experiment for this unit. Here’s a picture of the apparatus:

The quantities that the students manipulate are the hanging mass (external force), the radius of the mass on the “twister” and the mass on the “twister.” In this picture that mass is pretty small, later washers and an additional nut will be on each have of the “twister.” The lab groups have to determine how each of those quantities effects the angular acceleration of the system. Most groups measure the displacement of the falling mass and record that time to determine the linear acceleration, then use the radius of the “plastic wheel” (aka pulley) to determine the angular acceleration. What we eventually find at the end of the experiment is:

$\alpha \propto \dfrac{r_{pulley}\times F_{ext}}{m_{system}~ r_{masses}^2}$

After discussion, I reveal that the constant needed to make this an equation relates to the distribution of mass around the axis of rotation, thus we get:

$\alpha = \dfrac{r_{pulley} \times F_{ext}}{k~m_{system}~r_{masses}^2}$

To help relate this to Newton’s Second Law, we notice that in the numerator, is a quantity that increases rotation, and in the denominator, one that inhibits rotations. The quantity in the numerator we call torque, $\tau$; the quantity in the denominator we call the moment of inertia, $I$. Thus, we can rewrite the equation as:

$\alpha = \dfrac{\tau}{I}$

Before we move on, I make the point to explain that the multiplication to find torque is likely the first time my students have truly encountered multiplication with an $\times$. In Unit 6, we used the dot product to find work. That dot product of force and displacement meant that the two quantities had to be in the same direction, or parallel. Similarly, the cross product means that the quantities multiplied must be perpendicular.

From this point I we relate what we see here to what we saw at the beginning of Unit 3, namely to change the motion of an object we have to change the momentum of the object. Thus, we relate linear momentum, $p = mv$, to angular momentum, $L = I\omega$. We also relate translational kinetic energy, $K_T = \frac{1}{2}mv^2$ to rotational kinetic energy, $K_R = \frac{1}{2}I\omega^2$.

From there, we progress to worksheet 3, which, just like Unit 3, looks at momentum conservation of closed systems. In the worksheet, students are introduced to angular momentum diagrams. Since they have already seen the entire momentum diagram:

We jump to the complete angular momentum diagram. During this worksheet, we then have blank torque vs. time graphs.

The students use this diagram to help solve problems such as a what happens to the angular speed of a star when it collapses.

In worksheet 4, the students are introduced to Free Body Diagrams, in which they draw extended bodies instead of a small circle. I instruct the students that since we are no longer dealing with a particle, they need to draw the forces on the object where that force acts on that object. This worksheet then looks at systems in which the torques are balanced, which would constitute the traditional static torque problems.

In worksheet 5, the students now deal with conservation of momentum in open systems. Some of the problems also require the students to include kinematics relationships from worksheet 2 to solve problems.

In worksheet 6, the students are given first problem with a metal ball at the top of a ramp. The problem walks them through determining how long it would take the ball to go through a photogate at the bottom of the ramp without friction, and if friction causes the ball to roll without slipping. I then set up that ball on that ramp and we test their predictions. We usually get results within $\pm$0.0003 seconds of the time predicted for the ball rolling. We then do a similar problem for a solid disk and a hoop, with similar results.

From there, we review and take the unit test.

SWBAT:

1. determine and correctly use the graphical and mathematical relationships between angular position, angular velocity, and angular acceleration with time.
2. develop mathematical models from graphs of angular acceleration vs. torque and angular acceleration vs. mass, and angular acceleration vs. distribution of mass.
3. use rotational energy as an energy storage mode to solve problems using conservation of energy.
4. create Angular Momentum diagrams.
5. use conservation of angular momentum to solve problems.

## AP1 Unit 5: Unbalanced Forces Particle Model

The fifth unit, although not an introduction to the term impulse, is an in-depth study of momentum swaps between a system and its surrounding. The unit takes about three weeks. It begins with a Modified Atwood Machine activity.

The angle of the ramp is adjusted until the cart rolls down the ramp with zero acceleration. After a brief discussion of what is happening, what can be measured, and what can be manipulated, we settle into looking at how mass of the system and the magnitude of the unbalanced force are each related to the acceleration of the system. By moving mass from on the cart ($m_1$) to hanging from the string ($m_2$), the force can be adjusted without changing the mass of the cart-string-hanging mass system. Adding mass to the cart without changing the hanging mass, changes the mass of the system. By having a motion detector, the acceleration of the cart can be calculated for each run.

The data that is produced shows a proportional relationship between the net force and acceleration, and an inversely proportional relationship between the mass of the system and the acceleration. As you combine these two proportionalities you get:

$a \propto \dfrac {F_{Net}}{m}$

To write it as an equation, we must add a constant:

$a = k \dfrac{F_{Net}}{m}$

We then briefly discuss how for the SI System, this value of $k$ defines the derived unit $newton$ as the force required to accelerate 1 $kilogram$ at a rate of 1 $\dfrac{m}{s^2}$. While the value of $k$ for the English/American System is chosen such that the magnitude of the force is the same as the magnitude of the mass, and both are given the unit: $pound$.

We wrap up the post-lab discussion by looking at how the results from this activity change out momentum diagram. By including the impulse into or out of a system, we now have the complete momentum diagram we will use for the course. Here is an example for the diagram for a canon shooting a canon ball, treating the ball alone as the system.

Any object within the system goes inside the circle, any object in the surrounding that is involved in the event is written outside the circle. Every object inside the system is then displayed in the first and last bar graph. The width of the bar represents the mass of that object. The height corresponds to its velocity. The area of each bar represents the momentum of that object. The middle bar graph represents the impulse into or out of the system. The width corresponds to the elapsed time for the impulse. The height is the force. Thus the area of this graph is the impulse. For this level class, the shape tends to be a rectangle or a triangle. In a calculus based class, this could be any shape or function. Most of my students are familiar with Newton’s Second Law from their middle school science classes, but most haven’t seen the momentum diagram as a way to represent that equation.

From there, we progress to the model deployment phase of the unit. In worksheet 1, we look at elevator problems and other impulsive events. One of them features a momentum diagram with a trapezoidal impulse graph to challenge the students. In worksheet 2, we then combine the momentum diagram and kinematics from Unit 2. In worksheet 3, we begin to introduce 2-dimensional problems. The motion of the object is still in 1-dimension, but at least one force in the problem is in 2-dimensions. Students must find the components of that force in solving the problem.

After worksheet 3, we do the lab for this unit, which is the friction lab. Students must find the graphical and mathematical relationship between the normal force and the maximum static and constant velocity friction forces. Worksheet 4 then extends the problems seen in worksheet 3 to now include the resulting friction equations. We conclude the unit in worksheet 5. In this worksheet we work up to actual 2-dimensional projectile motion problems.

At the end of this unit, students should be able to:

1. create an interaction diagram including identification of the system and the types of interaction
2. use the relationship between normal force and friction correctly
3. create momentum diagrams that include impulses
4. solve problems using Newton’s 2nd Law correctly
5. apply Newton’s 2nd Law to predict the path of objects moving in two dimensions

## AP1 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.

## Momentum is King!

One idea that has been gnawing away at me is Andy Rundquist’s awesome notion that, “Momentum is King.” One quick link to see some of his ideas in greater detail is here. My issue is how exactly to make that stick in a modeling class, or stated another way, how does the storyline for mechanics change? The main issue I’m going back and forth over is if you teach momentum before forces, when and how exactly do you introduce the concept/term force?

Here are my thoughts so far:

I’m planning to progress though Constant Velocity Particle Model and Constant Acceleration Model as set up by the modeling content. (As a brief aside, if your not familiar with Kelly O’Sheas blog on modeling, go there ASAP!) From there, I’m planning to use a modified version of the Momentum Transfer Unit (Impulsive Force Model). After that is where I’m stuck, but here’s what I’m roughly thinking:

1) Jump to Balanced Forces, and don’t make the explicit connection between momentum transfer and forces. Basically begin creating a second parallel concept. Progress through the modeling materials for balanced forces (Free Particle Model is the official name) as designed and wait until unbalanced forces (Constant Force Model) to make the connection.

2) Recast the balanced forces model as multiple transfers model and remove the discussion of forces. Continue to reinforce the idea of momentum swaps, but now discuss the fact that the swaps can balance out. Focus on the system schema and stress that if there is not net momentum swap, the momentum must stay constant. Instead of hammering home “no net force, no acceleration; net force, acceleration” recast it through momentum. Then introduce the concept of force in the next unit, Unbalanced Forces Particle Model (which may or may not need/get a new name)

3) Jump to Unbalanced forces first and show the connection between impulse and Force. Begin by showing that a force is the rate of momentum change and that the area of a F vs t graph.  You could build the IF bar graphs in the momentum unit into IFF bar graphs. In the process focus on how each object has the same momentum swap, but different accelerations; that the forces are the same, but the effects may be different. (I’m looking at you bug on windshield problems!). Once complete, go to balanced forces and show that you can have multiple, simultaneous interactions, which can balance out. Now bring in things like system schemas and force diagrams.

My quick \$0.02:

1) If I’m making the effort to put momentum first, it doesn’t seem right to then ignore it when introducing the concept of forces.

2) Seems to flow a little better than (1), but when do you build in force diagrams? Adding them in the unbalanced forces unit make that unit really big, but I’m not sure I would feel right calling them momentum swap diagrams (especially if no other physicists do so).

3) Part of me is really drawn to this storyline, but in the end, that means I’m teaching Newton’s Law in the reverse order (3rd -> 2nd -> 1st). There is probably a good reason why no textbook or the modeling materials do it that way.

I’d love to hear feedback, if you’ve got any.

## Modeling Unit 9

Unit IX Momentum

(I can’t believe it’s really over!)
Jon mentioned that he does this unit a little differently, in that he has his students provide the definition of momentum on the Unit VIII test.  At the start of class he shows that list to the students.  What he has found is that most have a very good concept of momentum.  He said the modeling unit focuses more on changes in momentum (which tends to have more errors).  Usually from their definitions, he can lead them to the equation for momentum:

$\vec{p} = m\vec{v}$
or
$\Delta \vec{p} = m \Delta \vec{v}$

He said he also makes sure that they know that the units are $\left(kg \cdot m/s\right)$.  After being part of the Global Physics Department Meetings, Andy Rundquist, aka superfly, mentioned that he calls this “derived” unit a pom (particle of momentum), others at the meeting, name it after one of the students.  Jon mentioned that he names it after the first student that asks what is that unit called.

Next, Jon and Chris showed us the beginnings of collisions.  They attached a force probe to a ring stand at the end of a track.  They replaced the hook with a rubber bumper, and then had the extended spring end of the cart collide with the rubber bumper.  At the other end of the track they had a motion detector hooked up.  After zeroing and making sure that all probes were defined in the right direction, they had them collide.  On the projected screen, they had a plot of $F$ vs $t$ for the force probe data and a plot of $v$ vs $t$ for the motion detector.

They used the stats function on the $v$ vs $t$ plot to find the cart’s velocity before and after the collision (max and min values), and they multiplied these by the mass of the cart. (using the equations from the beginning of the unit $\large \vec{p} = m\vec{v}$.

Jon then walked/guided us through the derivation of Newton’s second law to show the relationship between Impulse (J) and Momentum

$F = ma$
$\large a = \frac{\Delta v}{\Delta t}$
$\large F = m \frac{\Delta v}{\Delta t}$
$F \Delta t = m \Delta v$

Jon then asked, “What is $p\Delta v$, to which we all replied momentum.  He said, well we call $F\Delta t$ Impulse.  He then asked, “What changes a velocity?”  To which we replied, “A force.”  He followed with, “What changes momentum?”  We answered, “Impulse.”  {If only all education was to people who already knew the material!}

Since the impulse changes the momentum, the magnitude of the change in momentum should be equal to the impulse.  Since impulse it force times time, we can find that quantity as the area under the $F$ vs $t$ plot.  Jon used the integration tool in LoggerPro, and amazingly enough, the value “matched” the change in momentum calculated from the $v$ vs $t$ plot.

We then jumped into Unit IX worksheet 1.

We agreed that #7 has some issues in that, for a rocket to go anywhere, it must lose mass.  Since we aren’t given that information, it technically can’t be solved.  However, Jon mentioned that we often start with idealized situations, and then add complexity.  We also agreed that most of our students wouldn’t know this anyway.

As we came back from lunch, we watch the PSSC video on Frames of Reference:
reference:

Frames of Reference

After that video, Jon and Chris showed us a cool video for E&M:

They next had a “student” come to the front of the room and sit on a stool, which was on a turntable.  They put a tennis ball in each of the student’s hands, and started gave the student a spin.  While spinning the student was told to release the ball so that it his a certain target.

Jon then thanked the student, removed the stool and got up onto the turntable himself.  He then had Chris throw a bowling ball to him.  After getting help to stop spinning, he threw the ball back to Chris.

From there, we moved into the actual paradigm lab.  We had a track with 2 carts.  Most groups had a small picket fence/flag to insert into the top of the carts.  Other groups just used a bent index card.  They also had two ringstands, each with a photogate attached.

Chris and Jon showed us several ways that the carts could combine, and we as a class agreed on 7 combinations we would study in our 7 groups.

1. 1 stationary cart, 1 moving with it’s spring plunger extended (between the two carts)
2. Both carts moving towards each other, one with plunger extended
3. One car moving towards the other, colliding with velcro between making carts stick
4. 1 moving cart, with magnetic repulsion causing the “collision”
5. Varying the mass of one cart, 1 cart moving w/ plunger out
6. varying mass of cart with both carts moving w/ plunger out
7. Both carts moving with velcro collision

From there we quickly ran through the pertinent parts of the paradigm lab discussion:
What can we measure?
Purpose:
To determine the graphical and mathematical relationships that exists between the total momentum of the system before and after a collision.

Right at the end of the day, Jon showed us a few more demonstrations.  First he hung a electrical tape “nest” from the ceiling.  Here are pictures:

Inside that cradle he placed a raw egg.  He set the length of the string to stop just before the floor, seen here:

Then, while standing on a stool, said to the students, think of this as you driving the car one day.  You happen to come around a bend in the road, texting away, and a tree decides to move itself into the road.  What happens if you are properly belted?  With that, he dropped the egg.  Since it’s in the nest, it bounces like a bungee jumper.  In his class, he then pulls another raw egg out of his pocket and says, this is what happens if you forget about your seat belt {drops egg -> splat!}.  Any questions?

Hey then gets 2 students to help him with his next demonstration.  He has one student help him hold a cotton table cloth as seen here:

If you look carefully, you’ll notice that they make a slight lip at the bottom of the sheet.  As the egg hits the sheet, they rotate it to horizontal, so that the egg won’t roll off.  Here’s an action shot of the egg hitting the sheet {quite impressive given that I was using an iPhone if I do say so myself}:

Lastly, Jon took out a tennis ball and the bowling ball (David recommended using a basketball to avoid damaging the floor, however, they didn’t have an inflated one handy).  Drop both from the same height, and you see that both return to about the same height.  Then, stack the tennis ball on top of the bowling ball and drop.  One word, Awesome!  Here are some pictures:

After that, FIU PER asked us to go into the hallway for a practice poster presentation of the research before they head off to the AAPT national meeting in a few weeks.  The couple things that jumped out to me {yes I’m probably butchering their edu-jargon terms, but I’ll give you the basic idea}:

• To great strategies for modeling are seeding and passive direction
• seeding: give one of the groups (especially struggling groups)  an important insight, so they have a key ingredient to share during the board meeting.
• passive direction: as the teacher, don’t be inside the circle (sitting w/students) if they don’t need you.  Allow them to take ownership of the meeting.  During the group work, determine where the misconceptions and errors are.  Let the groups work them out, only step in if they are floundering or off task.
• The guy had a third term he dropped, but I don’t remember it.  Basically he talked about learning what the students were doing, and planning you questions while they are working.  Give the class a chance to ask them, and add them in as necessary.
• Another poster talked about one powerful benefit of whiteboarding, namely that it allows students to interconnect with their peers, which improves their sense of belonging.  This improved attitude they have shown, had increased retention rates in the subject at the college level.  They speculate it would have an even more profound at the HS level.
• A third poster described how modeling allows for personal (mastery) interactions and more importantly “vicarious” interactions
• Their research has shown this is especially important for female students’ success in physics.

We start the last day with Unit IX worksheet 2 & worksheet 3

After finishing my work, I multi-tasked by looking at my twitter feed.  John Burk (@occam98) asked a great question while at a new teacher mentoring workshop:

To which I replied the concerns parents express with not “teaching” their child.  I’ve been using a lab based program (CPO Physics), and I’m guessing modeling teachers have similar issues.  I know I always have to go in to the idea that my job isn’t to tell the answer, but to find the best way to help their child learn the concept.  John replied that there is a lot of talk about this very issue in the modeling listserve.  For those that are thinking of moving into modeling, make sure you give a little thought to the question, “What is your job as a teacher?”  Is it to make sure you tell all the facts you expect the students to know, or is it to create an environment in which they can best learn your subject?  Personally, I hate when teachers talk about “covering” material.  I’ll get off my soapbox now.

We next went about whiteboarding our results to the worksheets.
Notes from board meeting

• Some of the problems need to be modernized, not sure if students would know what a “Geo” is, Cooper Mini or Smart Car might be better names for the small car.
• wkst 2 #7 needs to be cleaned up, give students names to avoid “former/latter” terminology

Lab Practicum
(def: looking for 1 final result not collection of data, using skills in the lab to now test the model)
Set up 2 carts 1 with known mass and 1 with unknown mass (tape masses to cart so they can’t be seen and can’t slide around) “stuck” together.

Use conservation of momentum to determine unknown cart’s mass- contest for either grade or some other prize

What worked?

• Egg seat belt demo
• All the other demos from Jon
• Designation of tasks in labs
• changing collision scenarios for each group
• PSSC Frame of Reference Video
• Having a practicum
• Jon breaks his class into 4 groups – all members must know how to do it
• Quiz the next day (small part of grade), only selects 1 persons quiz from each group for group grade
• Quiz is practicum calculations with slightly different numbers