Kevin Tran
Lab partners: Kevin Nguyen, Jose Rodriguez
May 27, 2017

Finding the moment of inertia of a uniform 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.

Introduction: We mount the triangle on a holder and disk. The upper disk floats on a cushion of air. A string is wrapped around a pulley on top of and attached to the disk and goes over a freely-rotating frictionless pulley to the hanging mass.

Below is the parallel axis theorem, which states that:

First, I like to note about the measurements within this experiment.

Top shell disk

-Diameter: 0.126 m

-mass: 1.356 kg

Bottom shell disk: 

-Diameter: 0.126 m

-mass: 1.343 kg

Aluminum disk

-Diameter: 0.126m

-mass: 0.467

Smaller pulley

-Diameter: 0.0248m

-mass: 0.0110kg

Larger puller

-Diameter:0.0498m

-mass: 0.036kg

Hanging mass

-mass: 0.025kg

Below is one of the three set ups that we compared.

With Loggers Pro, we were able to calculate the angular acceleration of the direction in the downwards and the upwards direction. The rising slope represents the downwards direction and the descending slope represents the upwards direction.

-The downwards angular acceleration for the system without a triangular piece is 5.775 rad/s^2

-The upwards angular acceleration for the system without a triangular piece is -6.994 rad/s^2
The average angular acceleration is 6.38 rad/s^2

Below is the calculated Inertia for the system without a triangle.

Next, below is the system with the triangle point pointing up.

Below are the calculation to solve for the center mass of the triangle









































Below is the inertia when the triangle is rotated around the edge and around the center of mass.





Below are the results of angular acceleration for the downwards and the upwards direction. 

-The downwards angular acceleration for the system without a triangular piece is 4.789 rad/s^2

-The upwards angular acceleration for the system without a triangular piece is -5.357 rad/s^2
-The average angular acceleration is 5.073 rad/s^2

Below is the calculated inertia for the system with the triangle tip facing up.

Next, below is the set up with the triangle turned 90 degrees.

Below are the results of angular acceleration for the downwards and the upwards direction. 

-The downwards angular acceleration for the system without a triangular piece is 3.698 rad/s^2

-The upwards angular acceleration for the system without a triangular piece is -4.142 rad/s^2
-The average angular acceleration is 3.92 rad/s^2

Below is the calculated inertia for the system with the triangle rotated 90 degrees.

We were able to find the difference of moment of inertia with no triangle with the triangle in each position. The moment of inertia of the triangle in a vertical position minus moment of inertia of the system with no triangle was 2.56 x 10^-4.

0.0012 - 0.000944= 2.56x10^-4 kg*m^2

We did the same for the triangular plate in the horizontal position. As a result, we found that its moment of inertia is 6.06 x 10^-4 kg*m^2

0.00155- 0.000944= 6.06 x 10^-4 kg*m^2

Conclusion: 

By finding the difference between the moment of inertia of the rotating system and then the moment of inertia of a triangular plate in two different positions, we were able to find the moments of inertia of the triangular plate itself. The values of these numbers represent the amount that it takes for the triangle(s) to spin around a fixed axis. The data allowed us to come to a conclusion that it takes a higher value for the triangle to spin around its center of mass in the horizontal position. The types of errors that could of affected our results are the air resistance of when the system is spinning and the mass of the triangles may not have been completely in uniform density.