14-May-2015: Moment of Inertia of a Uniform Triangle

Purpose:

To experimentally determine the moment of inertia of a right triangular this plate around its center of mass, for two perpendicular orientations of the triangle.
To then determine the "true" moment of inertia through calculations.
To compare our results.

Set-Up:

We mounted on a holder and disk onto the rotating system. The upper disk floated on a cushion of air to avoid friction. A string was wrapped around a disk attached to the upper disk and the triangle so that when the other end of it fell with a light mass, the string unraveled and turned the system providing torque. The torque pulley, disk and holder are shown below:

System without Triangle

We would then add the triangle onto the holder:

System with Triangle

Experiment:

1. First we set up the experiment so that we could find the angular acceleration of the rotating disk and holder when the light mass is dropped and provides a torque. Therefore, this will be used to find the inertia of the system without the triangle.

Angular Acceleration of Upper Steel Disk and Holder

2. Next we will need to collect the measurable data which included the mass of light mass, radius of the turning disk where the torque occurs, length, width, and weight of the triangle:

Mass of Light Hanging Mass

Radius of Disk with Torque

Width of Triangle

Length of Triangle

Mass of Triangle

Overall Recorded Measured Data: 

3. Next we performed the first step of the experiment so that we could find the angular acceleration of the rotating disk and holder with the triangle vertical when the light mass is dropped and provided a torque.

Angular Acceleration of Upper Steel Disk, Holder, and Vertical Triangle

4. Lastly, we recorded the angular acceleration with the triangle horizontal. 

Data Analysis: 

1. First we need to organize our angular accelerations. To find the angular acceleration we looked at the slope of the graph of angular velocity. Realizing that there is still slight friction between the steel disc even with an air cushion, we needed to take the average:

(rad/s/s)

2. Now that we have angular acceleration we need to know it's relationship to Inertia:

Torque = Inertia * Angular Acceleration

We are able to find the experimental Inertia of the Triangle by subtracting:

Inertia of System with Triangle - Inertia of System without Triangle = Inertia of Triangle

Using the same method from the previous experiment, we used our derived equation to find Inertia:

Next we are able to find our experiment's Inertia equation using the measured data:

mass of light mass = 24.5g (switch g to kg in the picture)
radius of disk with torque = 2.25cm

Experimental Inertia's found:

3. Lastly, we found "True" Inertia's of the vertical and horizontal triangle.

We first had to find the moment of inertia for a triangle being spun around it's edge:

Next we used Euler's law of motion to find the moment of inertia of a triangle rotated about its center of mass:

Using this equation we were able to plug in the vertical and horizontal data to find the "true" moment of Inertia:

Conclusion:

Comparing the results of the experimental inertias to the "true" calculated interias, we found this percent error:

This shows that our experimental values and calculated values were very close, a successful data analysis. Although uncertainties may have occurred with taking the measurable data or not getting a close enough angular acceleration reading, our results proved that moment of inertia can be found through correct integration, just like we have been doing in class, and reflect "real-world" experimental results.