# A Synthesis Of All The Models (Thus Far)

In this post I will describe a culminating activity for the first year students in the Academy. This is really the destination that the students have been headed towards since the beginning of the course. Everything they have learned is synthesized in this activity where the students gather data from various observations/experiments and then use the data to predict their own model rocket’s journey.

Note: There were two significant simplifications that we had to make based on the ability level of the students and the physics content covered in class. We had to assume that there was no air resistance force acting on the rocket during the thrust and cruise phases. We also assumed that the mass of the rocket did not change. I intend to have the students reflect on how this might affect the predictions and then analyze the actual performance data. More on this later…

# Measuring The Rocket Engine Thrust

We first needed to figure out the average force exerted by the rocket motor on the rockets and the time interval during which that force would be applied. This would give the students both the thrust force and the length of time of the thrust phase. We needed to collect force measurements for the rocket motors that we were using (C6-5). You can actually download this from many different websites, but it was much more fun to actually do it ourselves! Mr. Holt made a neat little rocket motor holder that was attached to a force meter and we went out into the rain to test the motor (see video below – thank’s Gary!):

The force data was then shared out to the students – here is what the graph looked like:

And this is the force vs time graph one retailer posted on their website:

Although the students had not been introduced to the concept of Impulse-Momentum transfer, we can use the average force, and that seems to work out really well. Just to make sure we could do this, I used the Integral tool in LoggerPro to measure the impulse, and it came out to 8.83 N s – really close to what Estes states – 8.8 N s.

# A Mini Wind Tunnel Test

The students then needed to measure the drag force on their parachutes (all cut on our new laser cutter) as a function of air speed so that they could estimate the terminal velocity of their rocket during the descent phase.  Next step was to test the parachutes. Luckily, Mr. Holt and I had helped two of our previous students create a really nice wind tunnel. We used a force meter attached to a vertical post inside the tunnel…

…and then we used a little Kestrel anemometer to measure the air speed…

Students were able to increase the air speed in the tunnel by turning a rheostat that controlled the fan speed. They then measured the wind velocity and graphed that against the measured force – just like NASA!

Here is some sample data to show how the results came out – not bad!

Students now had a way to estimate the descent velocity because they could calculate the gravitational force on their rocket, using the measured mass of their rockets, and then they could use their data to find the corresponding wind speed.

# Putting It All Together

As part of their final (50%), the students were asked to then take this data, measure the mass of their model rocket and construct a prediction. The prediction was to include these five elements:

1. A set of force diagrams for the different phases – thrust, cruise, and descent. The diagrams also had to include accompanying net force equations.
2. An acceleration vs time graph.
3. A velocity vs time graph.
4. A position vs time graph.
5. Finally a calculation sheet that includes all calculations required to create the motion graphs.

The students have been asked to turn this in before the actual launch.

As we collected the data above, I never explicitly reveal how the data should be used to make these predictions, but I do give them some guiding questions that orients them. They work with their partner’s on this report, but I warn them that they will both be held responsible for understanding the process of creating the prediction report.

# Testing the Predictions

Each student rocket will be equipped with a small altimeter (from Apogee Rocketry – love this thing!).

This altimeter records altitude data in 1/10 of a second intervals, and we have found it to be very accurate and reliable. We will be launching next week, so tune in soon for an exciting update on how the launches went!

# Investigating An Unbalanced Net Torque

We started by looking at the fact that a disk experiencing a net unbalanced torque also experienced a change in rotational or angular velocity. The students used a rotary motion sensor to measure the angular position and the angular velocity of a disk experiencing a constant torque and the students immediately recognized the similarity between a particle experiencing constant linear acceleration and a rigid body experiencing constant angular acceleration.

So, the natural next step in our investigation was to determine the causal relationship between a net unbalanced torque and the angular motion state of a rigid body. The students discussed how we might set up an investigation that would help us understand this relationship, and I helped guide them towards a final investigation design where we used a rotary motion sensor attached to the wooden disk we had used in a previous investigation.

The rotary sensors from Vernier are rather expensive, but also quite nice. They come with a plastic spindle with three different pulley radii. The only issue that I have with these is that they do not include a useful screw for attaching objects to the pulley – they expect you to purchase their completely over priced accessory kit. I don’t suggest this. Instead, you can create your own and then use a 6/32 screw to attach them to the pulley – just be careful not strip the threads!

Working off a similar investigation when studying the causal relationship between a net unbalanced force and linear acceleration, the students recognized that we needed to include a force sensor for measuring the applied force. The students then had LoggerPro plot the points for the calculated torque (using a calculated column) and the measured angular acceleration.  The data isn’t super clean, but its good enough for the students to conclude that the relationship is most likely linear.

# Dimensional Analysis – Inferring Rotational Inertia

I was amazed to discover that some students in the class set off to understand what the units of the slope could be reduced to. They immediately saw that the proportional constant (slope) included kg (mass) but that wasn’t all. After some work, the class had determined that the units for the slope were kg * m^2.

This led to a qualitative discussion about the inertia of rotating objects. We discussed hoops and disks primarily, and the class seemed to agree that the units made sense. Although I didn’t have any hoop-disk sets like the ones you can buy from various vendors, we did perform some thought experiments around mass distribution and rotation, but we needed to be sure that we were on the right track.

# A Better Variable Inertia Disk

So last year I was looking for an investigation that would really help the students discover the importance of mass distribution for a rotating object. Reading the material on the AMTA website regarding the unit on rotational motion, the researches stressed the importance of connecting mass distribution to the rotational inertia. I found this variable inertia disk from Fischer Scientific and decided to purchase it. I commend these guys on making this, but frankly I decided that I could make a better one.

My colleague and I set about redesigning these disks. The improvements we made included a) better compartments for the metal marbles so that they didn’t move around, b) an index and lip so that the two sides fit more securely together, and c) more compartments so that we can test more mass configurations. After we made our designs, we used our Makerbot 3D printer to print out ten copies of the disks.

With these disks attached to the rotary motion sensors, the students were able to confirm that although the total mass of the disk did not change, the angular acceleration declined as the mass was moved outward from the radius of rotation. This was more of a qualitative investigation, but I think the students were able to clearly see how moving the mass farther from the axis affected the rotational inertia of the disk.

# Deploying The Model (So far)

Now that the students had built a predictive model that described the quantitative relationship between rotational inertia, angular acceleration and net torque, they were ready to test it.

We returned to the investigation setup with the disk, but this time attached a hanging mass to the string and they attempted to predict the angular acceleration. The students immediately made the mistake (as I thought they might) in considering only the disk in the system. This gave me a chance to review with them the following process:

1. Identify the system schema (now it includes things that can rotate!)
2. Draw your force vector (free body/particle) diagrams and now also your force-moment arm (rigid body) diagrams.
3. Write the summation of forces AND the summation of torques for the system.
4. Do some algebra.

The trickiest part to figuring this out is getting the signs right due to interaction pairs and making sure that they agree. I ask the students to start with a diagram and then go through and label each force + or – by picking a force and then finding its partner and then making sure that the rest of the force directions agree.

Once they were able to get a prediction that we all agreed seemed correct, they ran the experiment. The class was able to predict the angular acceleration within about 10% error. We discussed the possible reason for discrepancy which led to some interesting discussions around modeling the rotational inertia of the attachment screw and washer, and also the frictional forces at work.

The students were pretty convinced that the model worked, and so next stop – energy and momentum in systems with rotational motion.