The Exploring Rotational Inertia Classroom Kit
When it comes to teaching rotation there are very few lab options. To address this need, we have developed the Exploring Rotational Inertia Classroom Kit. It provides exactly what AP Physics teachers, or any teachers who like hands-on lessons, need to deliver inquiry-style discovery labs to their students.
Figure 1. Students and teachers can set their own rotational inertia situations with these easy to manipulate, durable plastic disks.
Each kit contains two disks which can be assembled to contain eight total spheres. (Each disk comes with four spheres, but you can mix and match.) The plastic housing is easy to separate and reassemble but is snug enough so that the spheres do not rotate during motion. This is an essential design detail because in other kits, when the sphere moved about, it caused teachers all sorts of experimental errors and calculation blunders. Never again!
Figure 2. Here we see some of the equipment that comes with the classroom kits. The string allows for rotational collisions and free rotation. The long axle, center, allows for angular acceleration by torque labs, and has little O-rings to keep the disks in place.
By themselves the disks present several lab opportunities, and several other included accessories extend the activity possibilities. First is the string, which allows for rotation experiments, torque and angular momentum investigations. This can easily be tied through the center or wrapped around the disks. Next is the long axle, which enables you to permit rotation with very little friction. The black O-rings prevent slipping left and right on the low-friction axle. Lastly, there are the short axles, red and blue in the photo. These allow participants to really feel the inertia as they spin the disks. They are screwed snugly through the center of the disks, as I will discuss later.
Of course, the first thing you will want to do is predict the results of a downhill race. This classic experiment immediately demonstrates that – even though they might have the same mass – these two disks will not rotate with equal rapidity.
“Who will win? The center-weighted disk or the outer-weighted disk?”
Figure 3. A downhill race between equal mass disks will not result in equal descent speeds. The disk with the masses further out always loses. Apparently, there is more inertia in objects when their masses are further from the center.
This experiment presents a great opportunity for slow motion video. The disk with the masses further out always loses. This helps us to invent the formula for rotational inertia. But why does this happen?
Figure 4. The disk with the mass toward the center is going to win this race. It is easier to rotate this center-weighted disk because the masses have to be accelerated through a shorter distance per rotation.
Theory of Rotaional Inertia
The masses are going to be moving more quickly if they are rotating at a greater distance from the center of the disk. This means that they will have more kinetic energy, even if the masses are the same. Imagine a single rotation, if the masses are closer in, they will not have as high of a velocity. Thus, to induce rotation in a system, especially when the masses are further from the center, is to accelerate those masses at a higher velocity than if they were closer. We conclude that this takes more work, more effort, and ultimately feels more difficult to cause a rotation.
I = mR2
Feeling the Rotational Inertia
By spinning them at the same time, you immediately feel that the center weighted disk is easier to spin than the outer weighted disk. This is because it takes more work to move the one with the masses farther out. They acquire more kinetic energy, so this requires more energy from you to reverse their direction.
Figure 5. With the axles in place, it is easy to feel the rotational inertia of the disks. Your fingers definitely feel the sensation. Not only is it harder to get the outer-weighted disk into motion, it is also more difficult to stop it. The axles screw easily in place and can be assembled by teacher or student.
Calculating the rotational inertia it is a two-step process. We first have to measure the distance from the center of the disk to the center of the spheres. You will be interested to note that the outer spaces are twice as far as the inner spaces. This allows for easy calculation and comparison. However, we cannot deny that there is also the mass of the disks themselves, so these will need to be taken with a digital scale.
The formula for calculating the rotational inertia of an inner-weighted disk with four spheres is:
I = Io + 4*M r 2
Where Io is the inertia of the empty disk itself. This can be estimated by measuring m and then using the disk formula: Io = ½ m R2. This formula is true for any solid disk of uniform density. (Note here I am using R for the Radius of the disk and not the location of the masses.)
Even though this type of problem is extremely common on tests, it is extremely rare as a lab. The ability to examine rotational collisions is a great advantage of the rotational inertia classroom kit. In this experiment, one disk is dropping upon another. In this rotational collision, the disks will be as nearly identical as possible (although this is not essential for this experiment). The disk on the bottom will be spinning and hanging on the string and the top disk is dropped from a stationary state.
Figure 6. The author performs the rotational collision experiment.
When they finally reach the same speed, the pair is now rotating half as fast as the original rate. This is a demonstration of the conservation of angular momentum, which is very similar to the conservation of linear momentum. The total momentum is conserved before and after collision events. In this case, the inertia doubles and the rotational speed halves.
Figure 7. A close up of the experiment. After the top disk falls upon the lower disk, there will be a loss of angular speed by half. This is only because the inertia has doubled!
Figure 8. This type of angular collision is an extremely common question on the AP Physics exam. This is true despite how UNCOMMON this type of lab is. Well, not anymore!
The Torque Unwinding Experiment - a discrepant event
Figure 9. Shown above are the two mass configurations involved in the torque unwinding experiment. The surprise is that the less massive disk will have greater inertia.
To perform the torque unwinding experiment, wrap the string around each of the disks while their inertias are different. Any different arrangements will make good candidates for comparison. However, the comparison I have displayed in Figure 9 is perhaps the most interesting. Because the radius is twice as far, and that value is squared, it will have four times the effect on augmenting the rotational inertia.
Figure 10. By winding the string around the disks, it is possible to test their rotational inertias directly. The same applied torque will cause them to unwind at different rates.
The applied torque, though nearly identical, does not generate identical rotation rates. The less massive disk will actually unwind slower! This seems to defy Newton’s Laws – until you realize that rotational inertia depends on the square of the distance from the center of the disk. This experiment can also provide an opportunity to measure rotational inertia.
It is easy to predict and verify rotational inertias with a ruler and a digital balance. Of course, we calculate the rotational inertia with the above formula, but it is also possible to create and approximate certain situations. For example, I like to check how well the formula for a solid disk (I = ½ mR2 ) agrees with experiment. I realized that one loads every chamber with a sphere that it would approximate a solid disk. Thus, the time to roll down the hill can be predicted and checked. From this, and the dimensions of the disk, we can verify the formula as – for example – the disk rolls downhill for a measured time.
Figure 11. You are only limited by your imagination when coming up with interesting inertia possibilities.
The rotational inertia kit gives students a chance, not only to engage with inertia themselves, but to change it quickly and easily. This kit is unique and effective because it snuggly grips the metal marbles so they are not free to move. The fact that the disk snaps open to be manipulated and is durable and clear, speaks to the student-focused design of the kit. When you can see what is happening, you can understand what is happening. Rotational inertia is now a hands-on experiment.
James Lincoln is an experienced physics teacher with graduate degrees in education and applied physics. He has become known nationally as a physics education expert specializing in original demonstrations, the history of physics, and innovative hands-on instruction. The American Association of Physics Teachers and the Brown Foundation have funded his prior physics film series and SCAAPT's New Physics Teacher Workshops. Lincoln currently serves as the Chair of AAPT's Committee on Apparatus and has served as President of the Southern California Chapter of the AAPT, as a member of the California State Advisory for the Next Generation Science Standards, and as an AP Physics Exam Reader. He has also produced Videos Series for UCLA's Physics Demos Project, Arbor Scientific, eHow.com, About.com, and edX.org.