11-May-2015: Moment of Inertia and Frictional Torque Lab

The purpose of this lab was to find the moment of inertia of a disk and use that determine the frictional torque that acted on it. We would then use this information to determine to solve an inclined cart problem.

Part 1: Finding the Frictional Torque:
The apparatus consisted of a large metal disk attached on a central shaft. The disk would spin on this shaft (see Fig. 1).

Fig. 1

To begin, we needed to understand what we were exactly looking for. Considering that the net torque is equal to the moment of inertia multiplied by the angular acceleration, we needed to find the moment of inertia of the entire rotating object first. To do this we needed to split the apparatus into three components: two shaft cylinders, and the main center disk. With the equation for the moment of inertia of a disk being I=(1/2)MR^2, we needed to find the radius of all three of the disks. To do this, we used calipers to precisely measure their diameters (see Fig. 2) and then divided this value by 2. Once we had the radii, we needed to find the mass of each individual piece. To do this, we used percent composition. That is, we found the volume of the disk and two cylinders, totaled it up, and found the percentage that each piece made of the whole. To find the volume, we simply found the thickness of each piece (see Fig. 3) and used the equation V=pi*r^2*h.If we had the percent composition, we could then find the percent composition of mass for each piece. On the disk was labeled the entire mass of all three components together. We broke this mass into the percentages we calculated and got our mass for each piece (see Fig. 4 for calculations).

Fig. 2
Measure the diameter of the disks, we used calipers to get a precise value, and then took a picture of the calipers reading to analyze it more thoroughly.

Fig. 3
Using the end of the calipers, we were able to measure the lengths of the smaller disks.

Fig. 4-1
Here we found the percent composition of the volume of the rotating apparatus. 

Fig. 4-2
We used the percent composition to determine the mass of each individual piece of the rotating apparatus.

Once we had our masses, we found the moment of inertia of the two cylinders and center disk, and then summed it all up. Our final moment of inertia of the entire apparatus was calculated to be 0.019986 kg * m^2 (see Fig. 5 for calculations).

Fig. 5

Once we had our moment of inertia for the apparatus, we then proceeded to find the deceleration of the disk due to its frictional torque. To do this, we set up a camera at one end of  disk, directly lined up with the central shaft (see Fig. 6). Once we opened LoggerPro and ensured that we were getting live video feed, we gave the disk a gentle spin and let it slow down to a stop.

Fig. 6
We lined up the camera to get an accurate visual of the disk.

Once we had our video capture, we went back to LoggerPro to analyzed the data. To do this, we placed a coordinate system into the video, placing our origin directly over the central shaft. From there, we marked one specific point on the disk and followed it frame by frame as the disk slowed down (see Fig. 7) NOTE: We placed a black piece of tape on the edge of the disk to clearly show the point which we were marking in LoggerPro.

Fig. 7
By placing dots at the black edge, we were able to obtain x- and y-coordinates for the position of the edge.

Once we had our points marked, we found that we had x- and y-components of velocity throughout the circular motion. Thus, in order to find our tangential velocity, we had to take the square root of the velocity in the x-direction squared and the velocity in the y-direction squared. We defined a column in LoggerPro to do this for us at every point (see Fig. 8).

Fig. 8
Using LoggerPro, we were able to find the tangential velocity at every point.

When we plotted the tangential velocity versus time, we found that the velocity had a downward linear like trend. Thus, in order to find the tangential acceleration, we found a linear fit for the graph. (see Fig. 9). This value for acceleration could then be divided by the radius of large center disk and found our value for angular acceleration (see Fig. 10).

Fig. 9
By finding the slope of the tangential velocity graph, we are able to obtain the tangential acceleration.

Fig. 10

With all of our calculation completed, it was now time to plug into our net torque equation to find the frictional torque of the system. When we multiplied our moment of inertia and angular acceleration, we found the net frictional torque to be -0.01808 (kg*m^3)/s^2. 

Part 2: Testing Our Value:

To test our value for the frictional torque, we set up an inclined cart system. This cart system would consist of a cart that would be attached by string to one of the small cylinders on the rotating apparatus. The cart would be placed on an inclined track, where it would be release from the top and allowed to roll down. The goal was to make a prediction for the amount of time it would take the cart to go down the track and then test that value (see Fig. 11)

Fig. 11
The cart would slide down the track while we recorded the amount of time it takes to reach the bottom.

First, we had to find an expression for the amount of time it would take the cart to go down the track. To do this, we combined our net torque and net force equations. We found the mass of the cart using a scale and the angle of inclination using our phones. (see Fig. 12 for full calculations) Our prediction was 7.95 sec.

Fig. 12

When we ran the actual experiment, we obtained a consistent value of 8.64 sec. This yielded us a percent error of 7.99%. The fact that the actual value of time was larger indicated that there were some forces that we did not account for, which is reasonable.

Overall, we were able to justify our value for frictional torque with our inclined cart test. However, some possible sources of error include our measurements, our ability to properly place precise dots on our video capture, and our ability to press our stopwatches in a timely fashion.