13 November 2014: Triangle Moment of Inertia Lab 17

Objective:
The purpose of this lab is to determine the moment of inertia of a uniform triangle about its center of mass. We must also determine the angular acceleration of a hanging mass when the triangle is at different orientations:

Setup:

A triangle was given to each group. First the moment of inertia of the triangle must be found. But then, a the triangle must be placed onto a disk that is spinning freely without friction. A string is wrapped around a pulley and attached to a disk. This string then goes over another pulley and is attached to a hanging mass. There apparatus comes with a built in rotational sensor that can record angular velocity and position.









Procedure and Results:
The first thing to do was to gather data on the everything we were to use including measurements of the triangle and measurements on the disks upon which it would be placed:

Next, we had to calculate the moment of inertia of the triangle. A picture of our calculations can be seen below:

Since we had found the moment of inertia, it was time to test it using the apparatus shown above. First we found the angular acceleration of the system without a triangle. The graph can be seen below:

We must take the average of both the acceleration as the mass is going down and the acceleration as the mass is going up. This because there is some frictional torque in the system. So the average of both these values is (2.993+1.946)/2 = 2.469 rad/(s^2)

Next we placed the triangle on the disk pulley holder system and again found angular acceleration. We placed the triangle on the disk with the long side on the bottom.

Again we took the average of both values and found the average to be (2.564 + 1.470)/2= 2.017 rad/(s^2)

Then we rotated the triangle on the disk pulley holder system so that that the short side was on the bottom. We again found angular acceleration:

We took the average again and found it to be (3.013-1.670)/2 = 2.34 rad/(s^2).

We then took the time find the actual moments of inertia of the triangle:

Moment of Inertia of a Short Triangle

Moment of Inertia of a Tall Triangle

Next, using our average angular accelerations found above we were able to calculate the moment of inertia of each system:

With this equation for I, we simply plugged in values and found the moments of inertia of each system organized in the table below. It is important to note that after we found the moment of inertia of a system and the moment of inertia of the system with a triangle, we subtracted the value of the moment of inertia of the system from the value of the moment of inertia of the system and triangle to find the moment of inertia of just the triangle. In other words:

I(triangle+system)-I(system)=I(triangle)

Conclusion:
Our results prove that we were closer to calculating the moment of inertia of a short triangle than a tall triangle. Nevertheless, both percent errors fall within an acceptable range proving that we were correct in our calculations of moment of inertia of a triangle along its center of mass.