22-May-2017: Finding the moment of inertia of a uniform triangle about its center of mass

Lab 17: Finding the moment of inertia of a uniform triangle about its center of mass

22-May-2017
Purpose: Determine the moment of inertia of a right triangular plate around its center mass, upright, and on its side.

Summary:

Our set up here is identical to that of the angular acceleration lab's disk-hanging mass system, where we have a sensor at 200 counts per rotation. Thus we obtained graphs of angular position and angular velocity vs time, because with our angular velocity vs time graph, getting the upward and downward slopes of this graph  gives us our values of angular acceleration. The hanging mass exerts a torque on this system. Getting the aforementioned data would therefore let us find the inertia of the various systems.

1) Subsequently, we first record the upward, downward, and calculate the average of, the angular acceleration for the steel+aluminum disks+holder.

2) Then, we weigh, and measure the dimensions of our triangle plate, then place the triangle in an upright position with the triangle's center-mass placed directly into the holder. Then we record the upward, downward, and calculate the average of, the angular acceleration.

3) I repeat step 2, but rotate the triangle 90 degrees.

Knowing the angular acceleration of these systems lets us calculate for the inertia of each individual total systems (holder, upright triangle+holder, sideways triangle+holder), and because we can also derive the moment of inertia for the machine with just the holder, we can also derive the inertias of both triangle orientations by subtracting the inertia of the holder-system from our last two tests.

Analysis

The goal of this lab is to calculate the inertia of the triangles at upright and sideways orientations rotating about their center mass. To that effect, using the spinning disk-hanging mass system required us to find the inertia of the disks with the triangle-holder, the upright triangle+holder, and the sideways triangle+holder. Subtracting the inertia of the disks+holder from the last two cases lets us do this.

As aforementioned, our setup for finding the inertia is identical to our angular acceleration lab. In order to calculate the inertia, we measured the mass of the various parts of our apparatus, and used logger pro to find the angular acceleration, which was a matter of finding the slopes of logger pro's angular velocity vs time graphs as the hanging mass goes up and down. For our calculations, we obtain the average of these values. 

Afterwards, it's a matter of plugging in our values into our given equation of inertia, once again ignoring the friction that may be present in the disk, and in the pulley itself. This is so, for all three of my trials. These values represent the experimental part of the lab, because we want to be able to compare this to calculations for the same inertias, using derivations for the moment of inertia of a right triangle.

Angular accleration and dimensions of our triangle

Derivation of inertia for the three systems.

For our theoretical values for the moment of inertia, we use the equation for the moment of inertia of a right triangle that we calculated in the lecture videos. We first derived an expression for the center mass of the triangle, then the moment of inertia about the leftmost edge, and using the parallel axis theorem, solved for the moment of inertia about the triangle's center mass, and again, plug in our values from the experiment.

Doing that, we performed percent error calculations between the theoretical and experimental results for our triangles' center mass, and found them to be rather similar. Between our upright triangle, I got a percent error of 7%, and for the second, 4.7%, values, demonstrating a reasonable accuracy between my calculations and experiments. This is how we determined the moment of inertia of a right triangular plate around its center mass, upright, and on its side.

Calculating moment of inertia about the center mass of our triangle

Calculating our theoretical values for Icm using our derived expression.
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Final percent error calculations

Measured Data:

Graphs:

Angular acceleration of the holder system

Angular acceleration of the holder+upright triangle system

Angular acceleration of the holder+sideways triangle system

Calculated Data:

Conclusion:

Because of the small calculated percent errors for inertia, I conclude that my values for the inertia of the right triangles were reasonably valid.
Regardless, there were uncertainties in this lab that explains the percent errors that I got.
As usual, uncertainties like the scale wouldn't affect our calculations that much -- our most important source of error carries over from our angular acceleration lab - the friction present in the disk and the pulley.

Though the air pressure reduces the friction between our disks significantly, it is still present, both between the disks, and the string-pulley. In our angular acceleration lab, we ignored this friction in calculating inertia -- and we do so again, for our experimental values of the triangle's inertia. This means a value for our angular accelerations that are inaccurate, which creates uncertainty for all of our experimental inertia calculations.

Our base assumption that we can ignore friction is what most likely lead to the modest percent errors - as 7% is not insignificant - therefore, this assumption was clearly flawed, and if we had taken it into account, no doubt our final result for the theoretical would've been much smaller, and much closer to that of our experimental results.