This project allows you to have a educational introduction to the Moment of Inertia. The presented rolling cylinder has two chambers with equal volume. Having two cylinders: one filled the inner, the other filled the outer chamber, gives you two optically equal objects with totally different Moment of Inertia.
If the mass fo the rigid body is close to the rotation axis, the Moment of Inertia is rather small. Mass distributed distant to the rotations axis will lead to a rather big Moment of Inertia.
Let the cylinders roll down a inclined surface to present the unequal physical charactersitics. Since the Moment of Inertia determines the needed torque for a desired angular acceleration, the cylinder with the inner chamber filled will accelerate faster and reach the end of the inclined surface first.
Check my video showing a summary of the project:
Consinder the Rolling Racers animation, where all bodies do have the same mass:
There is no need for support rafts for printing the cylinder.
Printing the cap of the cylinder might require rafts. I printed the cap laying down flat with rafts in the closing circle slit and in the handle on top of the cap. In order to have a nice fitting of the cap and the cylinder, make sure to completly remove the rafts of the closing slit in the cap. Rasping and sanding leads to better fitting as well.
Make sure to print the cylinder and cap two times. So you can have a comparison of the two different Moments of Interia and therefore a race competition between to cylinders. This will make your experiment even more fun - for more information see project details below.
Since I'm teaching physics in highschool I know what it means to kids to experience physics in an experiment. Teaching stuff from books is fundamental and important. But the perfomance of experiments leads to better understanding and memories!
Because of the high costs of educational material, alot of school are not able to perform science experiments. Therefore I decided to design some things, which are easily printed and therefore low-cost-experiments.
Back in my schooldays I was fascinated of the experiment about the Moment of Inertia. Since there was no optical nor a weight difference in the bodys rolling down the inclined surface it looked like magic to me.
Thanks to this great memory I started designing this thing to provide a comprehensible experiment which allows everbody to understand the physical concept of Moment of Inertia.
I designed this by constraining the two volumes of the two chambers to be equal. Therefore filling the chambers up will lead to an equal weight!
Overview & Background:
By completing this project you will be able to have a hands-on experience and easy understanding about the Moment of Inertia. This project provides multiple options of usage:
The cylinder might act as:
- a hands-on DIY project for students
- a phenomenological introduction to the moment of inertia
- a qualitative lesson about the moment of inertia, including math
Students will experience the impact of the moment of inertia. By completing the project they will have a hands-on project to learn about the kinematics of rigid bodies.
Additionally this projects provides the opportunity to make some mathematical calculations about the moment of inertia.
Students at the age of 10-16 in physics or science.
This project is a physics / science project for high school classes.
Skills Learned (Standards):
Students building and interact with this project are going to learn:
- the basics of mechanics
- the physics of rotating rigid bodies
- the physics of mass distribution
How to build the Rolling Cylinder
You’ll need the following things:
- the 3d printed parts: cylinder + cap (2 times)
- some candles
- a (kitchen) scale
Follow the video (gallery) in order to build the Rolling Cylinder.
First you have to print the 3d parts. Make sure to print the cylinder + cap two times. After printing, safely remove the support rafts and use some sandpaper in order to have a nice fitting.
Once your 3d parts are printed you have to get some (old) candles. About 250g grams should be enough. Put them in a pot. Take another pot with boiling water to melt the candle wax. Don’t worry about the 3d printing material to melt. Since I did some tests and it always survived without having any scratch.
Once the wax is liquid, you have to safely fill it into the cylinders. Start with the inner-chamber-filled-cylinder first, since this one is easier. Make sure to not fill any wax into the outer chamber by protecting it with plastic wrap or something else.
Fill up the inner chamber till about 1cm under the edge of the inner chamber (see pictures). Now use a scale to figure out the mass of it. Place the second cylinder aka outer-chamber-filled-cylinder on the scale and start filling up the outer chamber. Stop filling it up as soon as you reach the same mass like the first cylinder.
The wax will take about 15 minutes to cool down. When cooling, down the volume of the wax will decrease. Therefore it will sink down a bit. But don’t worry, since the weight will be the same this is no problem for the later physical characteristics of the cylinder.
Now you should have to cylinders looking almost the same, having the same weight, but still, have different physical characteristics.
Let’s see about the physics in the next section:
In order to provide a good experiment, we need to take some heavy (high density) material. Therefore the physical aspects will appear even clearer. I think the easiest way is taking candle wax, but I did also think of gypsum or some kind of cement. These materials might all work well, you just have to be very cautious about the equal weight of the two cylinders.
Using 3d printed PLA, ABS or whatever will probably have too low density and therefore you might not be able to see any difference in the matter of moment of inertia.
Okay, just in case you are still wondering and did not yet google it:
What is the moment of inertia?
"The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body determines the torque needed for a desired angular acceleration about a rotational axis. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation." (Wikipedia)
Starting with the kinetic energy of a rotating rigid body, which is the sum of all mass points mi:
Applying the angular velocity:
Because all mass points are rotating around one axis with a given distance ri. So we get:
The bracketed part at the end of the equation is called moment of inertia:
For bodies with continuous mass distribution we have to integrate (instead of summation) over all mass points:
Knowing this will give us the opportunity to calculate the moment of inertia of several different bodies. Let’s start with an easy one:
Calculate the moment of inertia of a ring
Assuming a ring with the mass mr and the radius rr. The rotation axis is perpendicular to the surface of the ring and through the middle point of the ring. The entire mass is concentrated in the distance rr. Therefore we get a moment of inertia of:
Calculate the moment of inertia of a disk
The next step is calculating the moment of inertia of a flat disk with radius rd and the mass md. Again the rotation axis is perpendicular to the surface of the disk and through the middle point of the disk. It should appear clear that the moment of inertia of this disk is smaller than the one of the ring, since the mass is distributed from r=0 and r=rd. Therefore more mass is closer to the rotation axis than it is for the ring. Imaging the disk as a summation of multiple mass-element-rings. The area of one mass element is:
Since the mass is distributed homogeneous we find the area density to be:
With this we find one mass element to be:
Putting all this together, we are able to calculate the moment of inertia of the disk:
As expected the moment of inertia of the disk is smaller than the one of the ring.
Calculate the moment of inertia of a cylinder
In order to apply the calculation to our purpose we have to figure out the moment of inertia of a cylinder. Therefore we assume a cylinder with the mass mc and the radius rc. Imaging the cylinder as a summation of multiple disks with the masses dm and the moment of inertia of (as calculated above):
Integrating over all disks will give us the moment of inertia of the cylinder, which is:
Now we should be able to calculate the moment of inertia of the cylinders we are using. But bevor doing so I would like to introduce another way of calculating the moment of inertia of a cylinder, which is by using the cylindrical coordinates:
Using cylindrical coordinates
Depending of the math skills of your students you might be able to introduce the cylindrical coordinates to them. Therefore the moment of inertia is a very good example.
Forget about the ring and the disk, since we now want to start from scratch. By using cylindrical coordinates, it is helpful to integrate of the volume of the cylinder with multiple integrals. As you might know the volume of a cylinder is:
The density is defined as:
Together we get an equation for the mass of the cylinder:
Ok, let’s try to figure out the moment of intertia:
Since the mass distribution is assumed homogeneous we might omit the dependency of the density to r. We’ll now integrate three times: First the height of the cylinder, then the angle of the bottom area, last the radius of the bottom circle:
Don’t you think this was much easier? I think so, too!
Ok, since I want to leave some room for other educators I will stop doing math now. Just to give you some ideas of what else you could do with this project in your class, consider the following list:
- calculate the explicit moment of inertia for the two cylinders
- calculate the estimated times the cylinders take to roll down any inclined surface (tip: using potential and kinetic energy)
- let the students try to figure out the biggest and the lowest moment of inertia of a fictive body
As you can see there are a lot of applications to this project. Maybe you’ll have some other interesting ideas of interacting with this project. I would appreciate if you could share it, in order to improve the project and have a more awesome experiment students can interact and learn with.
Strongly depends on chosen content for teaching:
Just presenting the rolling cylinders as a phenomenological experiment: about 0.5h
Presenting the rolling cylinders, discuss about the physics and have some calculations about the moment of inertia: about 2h
- Presenting the rolling cylinders, discuss about the physics and have a lot of calculations including the moment of inertia calculation with cylindrical coordinates and the theoretical estimation of the needed time the cylinders will take to roll down an inclined surface: about 4h
Students should know about:
- physics of kinematics of rigid bodies
- integral calculations
Rubric & Assessment:
At the end of the project, students should have:
- (optional: printed and successfully built the rolling cylinders)
- observed and understood the physical aspects of the moment of inertia
- made some calculations about the moment of inertia and therefore
- learned the math about the moment of inertia
- (optional: learned cylindrical coordinates)
Grading and assessment of the students:
- the overall behavior in this project
- did they manage to figure out the moment of inertia of the two cylinders?
- did they manage to calculate the moment of inertia using cylindrical coordinates?
- did they manage to calculate the estimated time the cylinders take to roll down the inclined surface?
- How do we have to imagine the mass distribution of a body with the smallest (/biggest) moment of inertia?