13-May-2015: Finding The Moment of Inertia of a Uniform Triangle about its Center of Mass

Purpose- We will figure out the moment of inertia about the center of mass of a right triangular thin plate, for two perpendicular orientations of the triangle.


For this lab we will use a new tool in determining the moment of inertia while rotating not at the center. This is called Parallel Axis Theorem. The moment of inertia of any object about an axis through its center of mass is the minimum moment of inertia for an axis in that direction in space. The moment of inertia about any axis parallel to that axis through the center of mass is given by



The expression added to the center of mass moment of inertia will be recognized as the moment of inertia of a point mass - the moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at the center of mass.



The experiment is as followed. We are using the same equipment from the angular acceleration lab where we have the Pasco rotational sensor with its disks, pulley, and hanging mass. This time we have a thin right triangle piece that will fit on top of the disk. Fortunately there is only one hole and it happens to be at the center of mass of the triangle. This will only allow us to run a set with the triangle long side vertical and another with the long side horizontal.

By winding up the string and hanging mass we let the system going and logger pro will assist us in developing data so that we may find angular acceleration. We will do three different trials three times to get an average angular acceleration of each system.


First trail is with disk and the holder. The slope of angular velocity give us angular acceleration. Though we must note that the values as the mass falls and rise are different. Which make sense since more mass is around the disk versus the hanging mass. Taking averages of multiply trials will help yield an accurate value for angular acceleration for each system.




Second is with disk, holder, and the thin right triangle piece positioned long side vertical.




Third is with disk, holder, and the thin right triangle piece positioned long side horizontal.

To reiterate the linear fit on the left is the mass rising back up and the linear fit on the right is the mass falling. During this yo yo effect the value of it rising is decelerating and falling going in the positive direction. If you look carefully at the angular velocity graphs, say were to allow logger pro to continue graphing the line. That line would get smaller and smaller until the moment of the system finally stops. The higher value of deceleration you have to take into fact that gravity plays a role.

Now we have these experimental values. Lets take a step back and derive the moment of inertia for a right triangle with the axis at the edge. 

We calculated inertia at the edge and we can take the parallel axis theorem part over so we can compare these calculations with moment of inertia center of mass from the experimental values.

Using the same moment of inertia equation from the angular acceleration lab. We will calculate "I" for both when the right triangle is long side vertical and long side horizontal. Which end up being identical in value. I, CoM =(2.408 * 10^-4) kg m^2

Comparing the both give us an error of 3.68 percent.


Error to take into account are, simulated zero friction is not completely zero friction, air resistance, and equipment seems like a go to but it would seem the only thing involving equipment would be how the air hose is hook up into the rotational device. The amount of air used and turning on and off the air can give different simulated friction free scenario in each trial. To conclude the lab I say it was a success in utilizing the new tool Parallel Axis Theorem to account for the distance the axis of rotation changes from the center of mass.