*Introduction to the Moment of Inertia Challenge*

We are going to assume that you have studied the concepts of *moment of inertia* and *physical pendulums* in your physics class. With that in mind, we present a "Moment of Inertia Challenge" for you in this lab. As you know, moment of inertia depends not only on the mass of an object, but also on how the mass is distributed, as well as the specific axis upon which it rotates. It is of particular interest to compare the moments of inertia of two objects with the *same mass* but having the mass *distributed differently*. You will be constructing physical pendulums like those shown in Figure 1.

You will be provided with the card stock white disks that have a pattern printed on them and 16 wood blocks of nearly the same mass each. These physical pendulums are hanging from an axis near the top edge of the disks. In the image on the left, the 16 blocks have been taped (using double-stick tape) to the blue squares--8 squares on the front and 8 squares on the back. In the image on the right, the 16 blocks have been taped to the red squares--again with 8 squares on the front and 8 squares on the back. *The result is two physical pendulums with the same mass, but with the mass distributed closer to the center of mass in one of the pendulums.* A tiny magnet has been taped to the bottom of each pendulum. PocketLab's magnetic field magnitude is recorded as the pendulum swings back and forth. This recording allows you to accurately calculate the period of the pendulum.

Figure 2 shows the detail printed on the white disk. The wood blocks are distributed evenly around the circles (45 degrees apart). A hole is punched in the disk very close to the top edge for the axis of rotation. A tiny magnet is taped to the bottom edge

The short combined data/video below from the PocketLab app shows what you can expect to see when performing the experiment.

*Your Challenge*

The moment of inertia challenge is as follows:

- Use PocketLab to collect and then analyze data allowing you to determine the period
*T*for each of the two physical pendulums. From the period and other measurable parameters, calculate what we will refer to as the "experimental value" for the moment of inertia in kg-m*m of each physical pendulum*about the axis near the edge of the disk*. - Derive an equation for the
*theoretical*value for the moment of inertia. Be sure to state any assumptions you are making. Then, using this equation, determine the theoretical value in kg-m*m.*about the center of mass* - Derive an equation for the
*theoretical*value for the moment of inertia. Then, using this equation, determine the theoretical value in kg-m*m.*about the axis that is very close to the top of the physical pendulums* - Determine the percent difference between the experimental and theoretical moments of inertia about the pivot axis
*.*Comparisons should be expressed as the percent difference between your experimental and theoretical moment of inertia about the axes near the edges, based upon a fraction of the theoretical moment of inertia. The percent difference would be negative in the event that your experimental result is less than the theoretical, and positive if the experimental result if greater than the theoretical.