13-May-2015: Finding the moment of inertia of a uniform triangle about its center of mass

Lab Partner: Jenna Tanimoto

PURPOSE

The purpose of this experiment was to find the moment of inertia of an object through experimental methods.

PROCEDURES

Figure 1

We began this experiment by constructing the set-up as shown in Figure 1. The set-up was very similar to the set-up used in Lab 16: Angular acceleration, except for the pulley connected to the disks. This pulley had an attachment that allowed us to place an object that would rotate with the disks. We first ran the experiment without anything added to find the moment of inertia of the system without the triangle. We would later subtract this from the moment of inertia found in the next two parts to find the moment of inertia of the triangle by itself.

Figure 2

Figure 2 shows the resulting graphs. From the second graph in Figure 2, we were able to find the angular acceleration of the system by looking at the slopes of the graph. We assumed that the parts of the graph that were sloping down represented the system when the hanging mass was going up and vice versa for the parts of the graph that were sloping up.

Figure 3

Next, we added a right triangular plate to the system by attaching it to the pulley at its center of mass, as shown in Figure 3. We first had the longer side as the height and the shorter side as the base. Then in the second part of the third part of the experiment, we changed the orientation of the triangle so its base was bigger than its height.

Figure 4: Triangle with h > b

Figure 5: Triangle with h < b

Figures 4 and 5 represent the disks' motion with the triangle in the two different orientations. We found the angular acceleration of the disk for each trial using the same process as the first part of the experiment (from the slopes of the graphs). All the values that we found are summed up in Figure 6 below. 

Figure 6

From the values on the left side of Figure 6, we found the average angular acceleration for the three cases (no triangle, triangle h>b, triangle h<b), which were 6.640 rad/s², 5.266 rad/s², and 4.192 rad/s², respectively. The dimensions of the triangle and the distance between the y-axis and the center of mass of each triangle are also shown in Figure 6.

Figure 7

Figure 7

After finding the average angular acceleration of each case, we were able to derive the moment of inertia using the same equation that we used in Lab 16 (Figure 7). The r in the equation refers to the radius of the pulley, while m refers to the hanging mass. The α represents the average angular acceleration of the disk. We found the moment of inertia of the system without the triangle (I = 0.0009078 kg*m²) to subtract it from the moment of inertia with the triangle (I = 0.001149 kg*m² for h>b and I = 0.001447 kg*m² for h<b) to find the moment of inertia of the triangle by itself. This process is shown in Figure 8 with the triangle in the different orientations. The Isys represents the moment of inertia of the entire system including the triangle and Io represents the moment of inertia of the rotating system without the triangle. For the triangle with its height bigger than its base, the moment of inertia was 0.00024 kg*m². On the other hand, the triangle with its base greater than its height had a moment of inertia of 0.000539 kg*m². This makes sense that the triangle with its base greater than its height had a bigger moment of inertia because more of its mass is concentrated farther away from the axis of rotation.

Figure 9

Figure 10

After finding our experimental values, we used calculus to derive a formula to find the moment of inertia of a triangle rotating about the y-axis as shown in Figure 9. We ended up with the formula I = 1/6MB². Then, we found the moment of inertia about the triangle's center of mass (for both cases) by using the parallel-axis theorem as demonstrated in Figure 10. The d is the horizontal distance between the y-axis and the triangle's center of mass.. These distances are illustrated on the bottom right corner of Figure 6.

Finally, we compared these values to the experimental values displayed in Figure 8. For the triangle with h>b, we got an theoretical value of 0.0002439 kg*m² and an experimental value of 0.00024 kg*m². The percent error between these values is 4.167 percent. For the triangle with h<b, we got a theoretical value of 0.0006047 kg*m² and an experimental value of 0.000539 kg*m². The percent difference between the expected and experimental values was 10.86 percent. The percent error for the second case was slightly higher that we would have liked.

CONCLUSION

In this experiment, we were able to get some reliable results. However, values found for the triangle with h<b was not as close as to the theoretical values as we would have liked. With a percent error of 10.86 percent, the error was not alarmingly high, but it was too high to just overlook. We believe that there could have been a few factors that contributed to this slight error. First, the hanging mass could have weighed more than we had assumed that it weighed based on the value marked on the mass itself. This would explain why the experimental moment of inertia was smaller than the theoretical one because the angular acceleration of the system was higher than it was supposed to be. One factor that we would have expected to contribute to the error in the experiment was the negative torque caused on the system by friction. However, this seems not to be our case because this would have made the experimental value being higher than the theoretical value because the angular acceleration would have been slower than we expected.