5-13-2015: Finding the moment of inertia of a uniform triangle about its center of mass

Lab 17: Finding moment of inertia of a triangle

Purpose: To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

Apparatus:

The apparatus shown above is the same apparatus we used for the Angular Acceleration Lab. For this lab, we used the same steel disks and large torque pulley as in the previous lab. The only difference being that this time we added a triangular plate to the rotating mass centered about its center of mass. A holder was added to the rotating stand to hold the triangle in place. Lastly, the same hanging mass was used in this experiment as in the Angular Acceleration Lab.

Furthermore, we adjusted the settings in LoggerPro so that it would record the rotary motion of the system and we also set the sensor settings to 200 counts per revolution.

The mass of the disks, torque pulley, hanging mass, and triangle were measured .
The diameters of the torque pulley and disks were measured.
The dimensions of the triangle were also measured.

Procedure:
First, we derived the moment of inertia of a uniform triangle about its center of mass. We did so, using the parallel axis theorem. This method is faster and easier because the limits of integration are easier if we calculate the moment of inertia around a vertical end of the triangle and then use the parallel axis theorem to find the inertia around the triangle's center of mass.

Inertia around the  vertical edge

Inertia around the center of mass

 Measurements of Disks and Torque Pulley and Triangle:

• Top Steel Disk: d=12.630 cm  m=1355g
• Bottom Steel Disk: d=12.630 cm  m=1348g
• Larger Torque Pulley: d=5.36 cm  m=36.3g
• Hanging mass: m=24.6
• Triangle: m=453.3g

Like in the Angular Acceleration Lab, we will run the experiment and record the angular acceleration of the rotating mass. Then, we will use the same derived equation that we used before to find the inertia of the system. However, this time we only need the inertia of the triangle; therefore we will run an experiment without the triangle to find the moment of inertia of the disk. Then, we will run a separate experiment with the triangle in place. Finally, we could find the inertia of the triangle around its center of mass by taking the difference between the two values. The last step is to compare the experimental and theoretical values for the moment of inertia of the triangle around its center of mass. This process was used to find the Inertia of the triangle in two orientations:

• Orientation 1: Height=14.83 cm, Base=9.85 cm
• Orientation 2: Height=9.85 cm. Base=14.83 cm

For this lab, we setup the apparatus so that only the top disk would rotate. Like before, we would turn the air on. Then we wrapped the string of the hanging mass around the torque pulley so that the hanging mass was at its highest point. Here, we would start the measurements and then release the mass. Next, we used the angular velocity graphs to measure the angular acceleration as the hanging mass moved up and down. The angular acceleration was found using the slope of the angular velocity graphs.

In the picture above, the positive slope is the mass going down and the negative slope is the mass going up. Using both angular accelerations, an average angular acceleration was calculated. This process was repeated for the triangle in both orientations. We used this method to find the alphas listed below:

• Alpha Avg for disk w/o triangle = 2.17 rad/sec^2
• Alpha Avg for disk w/Triangle in Orientation1= 2.005 rad/sec^2
• Alpha Avg for disk w/Triangle in Orientation2= 1.826 rad/sec^2

Orientation 1:
The picture below shows the derived equation we used to find the inertia of the rotating mass.

•  Inertia of the disk w/o triangle is 0.00296 kg*m^2.
• Inertia with the triangle is 0.00321 kg*m^2.

If we take the difference of these values, we find that the Inertia of the Triangle around its center of mass is 0.00025 kg*m^2

Next, we found the theoretical value for the moment of inertia using:
(1/18)Mb^2, m=0.4533 kg and b=.0985 m.

• Theoretical Value= .000244 kg*m^2

If we compare the experimental and theoretical values, we find that there is a 4% Error

Orientation 2: Here, we followed the same process shown above.

 In the picture above we show that in this new orientation, the equation for Inertia around cm is still the same. In this case, mass still equals 453.3g but the base is now 0.1483 cm.

• Theoretical Value of Inertia around cm = 0.000554 kg*m^2

Alpha for disk w/ Triangle

• Inertia of the disk w/o triangle is 0.00296 kg*m^2.
• Inertia with the triangle is 0.00354 kg*m^2.

To find the inertia of the triangle around its cm, we still take the difference between the inertia of disk and the inertia of the disk w/ triangle.

• Experimental Value of Inertia around cm= 0.000564 kg*m^2

If we compare the experimental and theoretical values, we find that there is a 1.85% Error.

Conclusion:
The theoretical and experimental values for the moments of inertia around the triangles center of mass for both orientations are very similar. In the first orientation, where the shorter leg is parallel to the floor, the percentage error is 4%. In the second orientation, where the longer leg is parallel to the floor, the percentage error is only 1.85%. This shows us that the method upon acquiring the Inertia around the cm is accurate. The very small percentage errors can be attributed to the uncertainty in the measurements of the triangle, torque pulley, and disks. Furthermore, error can also be found in the different calculated angular accelerations of the experiment.