14-Nov-2016: Lab 17 Finding the moment of inertia of a uniform triangle about its center of mass

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

    Name: Qiwen Ye (Sherry)

    Lab Partners: Xavier

    Date: 14-Nov-2016

2. Purpose

In this experiment, we determined the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientation of the triangle by using the parallel axis theorem. `

3. Theory

The parallel axis theorem states that

Because the limits of integration were simpler if we calculated the moment of inertia around a vertical end of the triangle, we could calculate that moment of inertia and then get Icm from:

Then, we used an equation (like above shown) derived from the previous lab to find the moment of inertia in each system. This equation uses hanging mass, radius of pulley, and average of acceleration.

4. Apparatus/Experimental Procedure
We mounted a holder and disk, and the upper disk floated on a cushion of air. A string was wrapped around a pulley on top of an attached to the disk and goes over a freely-rotating "frictionless" pulley to a hanging mass. The tension in the string exerted a torque on the pulley-disk combination.

Then, set up Logger Pro in Rotary Motion by 200 counts per rotation. Recorded the angular acceleration of the system in order to determine the moment of inertia of the system.

After we measured the moment of inertia of the system without triangle, we could mount the triangle onto the disk-pulley-holder system and measured its acceleration again and determined the moment of inertia of the triangle in vertical and horizontal orientation.

Vertical Triangle

Horizontal Triangle

5. Measured Date/Graph
hanging mass: m=0.0245kg
radius of pulley: r=0.0141m
the mass of triangle: M=0.4568kg
base of triangle: B=0.0987m ; H=0.1485m

Those three graphs we got from Logger Pro to find the average acceleration in each system. The slope of graph was its acceleration.

No Triangle: velocity vs. time

Vertical Triangle: velocity vs. time

Horizontal Triangle: velocity vs. time

6. Calculated Results
Through the above graphs, we got the acceleration in each system.

Here was the example, how could we calculated the average acceleration:

After we calculated the average acceleration in each system, we could calculate the moment of inertia in each system by using the equation we derived from the previous lab, and we used the moment of inertia of triangle system minus the moment of inertia without triangle to find the moment of inertia of the triangle in each of the two orientations. 

We could also calculate the theoretical value of the moment of triangle in each of two orientations by using parallel axis theorem:

7. Explanation

After we got the value of theoretical and experimental of the moment of the triangle in vertical and horizontal, we compared both values by finding the difference percent.

8. Conclusion

In this experiment, we studies how to use the parallel axis theorem to find the moment of inertia and how to use the torque to find the moment of inertia of an object. For the moment of inertia of the triangle, we compared the theoretical and experimental values, it shows that the difference of uncertainty was 18.12% in vertical triangle; 3.26% in horizontal triangle. Even though the horizontal triangle have a small different percent than the vertical one, our experiment still successful. The factors cause our theoretical and experimental values different are: the mass of the triangle plate may not completely in uniform density; the air being released to spin the rotating disks may have fluctuated that will cause the acceleration fluctuating; or our measurements a little bit error.