## Rotational Motion: Moment of Inertia

*Rotational Motion and Moment of Inertia Lab Setup*

Figure 1 shows a ramp and three distinctly different objects that you will release from rest at the top. Each object will roll downward to the end of the ramp without slipping, resulting in rotational motion. The roll of Gorilla tape has a shape known as an annular cylinder. The can of jellied cranberry sauce is a solid cylinder. The cardboard tube, in contrast to the can, is hollow. All three of these objects will rotate about their central cylinder axis while rolling down the ramp. Each of these three objects has a different moment of inertia when rotating about its central cylindrical axis. This results in different speeds for the center-of-mass of each object upon reaching the bottom of the ramp. It is important that the solid can be as solid as possible. Cans of soup or other liquids which slosh around as the can rolls do not meet the definition of solid.

A quick sample video of the Gorilla tape rolling down the ramp appears below. You will note that PocketLab Voyager has been taped to a piece of cardboard that has, in turn, been taped to the back of the Gorilla tape.

*Rotational Motion Student Challenges*

- Design a PocketLab-based experiment to determine the speed of the center of mass when each of the objects reaches the bottom of the ramp. Which app do you feel would allow for the easiest and quickest analysis -- the PocketLab app or the VelocityLab app?
- Specify any important assumptions you are making for data collected in your lab setup.
- Using conservation of mechanical energy, derive equations for the speed of the center of mass of the solid can and the hollow tube when they reach the bottom of the ramp. You will need to do a Web search for the moment of inertia of a solid cylinder and hollow cylinder about their central axis. According to the theory, is the speed dependent upon the radius and mass of these two objects?
- How do the theoretical speeds of the can and hollow tube compare to those you obtained from your experiment? What are some possible reasons for differences between your experimental and theoretical results?
- In a manner similar to that from challenge 2, obtain a formula for the theoretical speed of the center of mass of the Gorilla tape when it reaches the bottom of the ramp. Does this speed depend upon knowing the exact value of the inner and outer radius, or simply on knowing the ratio of these radii? Your equation for the Gorilla tape should reduce to the theoretical equation that you obtained in challenge 2 for the speed of the solid can if the inner radius is zero. Your equation for the Gorilla tape should reduce to the theoretical equation you obtained in challenge 2 for the speed of the hollow cylinder if the inner radius is equal to the outer radius.
- How does the theoretical speed of the Gorilla tape roll compare to those you obtained from your experiment? What are some possible reasons for differences between your experimental and theoretical results?

*Three Additional Lessons Dealing with Moment of Inertia*

The author has created three previous lessons in which students investigate the concept of moment of inertia. Here are the links for anyone interested in checking them out:

Rotational Dynamics of a Falling Meter Stick

PocketLab Voyager: Moment of Inertia and Conservation of Angular Momentum