4-Nov-2014: Moment of Inertia

Purpose:
The purpose of this lab is to find the moment of inertia of a large metal disk attached with two smaller cylinders, which spins together. Also, to find the time it takes to travel a certain amount of revolutions.

Apparatus:
There are two parts to this lab. First part is finding the moment of inertia of the disk and cylinders. To do that we find the volume of the disk and cylinders and the mass of the whole object. We calculated the moment of inertia by timing how long it takes to make a certain amount of revolutions before coming to a stop. Then, we used certain equations to find the torque for friction. With the torque found, we can move on to the second part of the lab. We placed a track at an angle of 36 degree above the horizontal and the apparatus at the top of the track. With one end of a string tied to one of the smaller cylinders of the apparatus and the other end a cart, we can release the cart and find the time it takes to travel a meter down the track. The time it takes should be the time we predicted, if not then close to that time.

Explanation:
First we calculated the moment of inertia of the disk and cylinder, and to do that we found the volume by measuring the dimensions of the disk and cylinder. The disk had a radius of 0.1 m and a height of  0.0157 m; the cylinders had a radius of 0.0157 m and a height of 0.0571 m. The whole object had a mass of 4.808 kg, so the moment of inertia of the disk came out to be 0.018 kg*m².

Finding inertia

After finding inertia, we measured the time for the disk to stop, and found the angular acceleration with the equation in the image below. The average angular acceleration was 0.493 rad/s², and with our inertia calculated earlier, we found the torque for friction to be 0.008874 kg*m². 

Finding angular acceleration

Finding torque for friction

To find how long it would take for the cart to travel a meter when attached to the spinning cylinder, we need to find acceleration of the whole system, cart and cylinder together. We solved for acceleration symbolically and came up with an acceleration of 0.0258 m/s², and 0.0264 m/s². We used kinetics to solve for time and came up with   8.76 secs and 8.56 secs. The actual time was 8.69 secs, which gave us a percent error of 1.5% and 0.8%.

Conclusion:
We found moment of inertia and calculate angular deceleration to calculate the frictional torque. With the frictional torque, we can calculate the acceleration of the whole system, cart and cylinder. Using the acceleration of the whole system and kinetics, we found the theoretical time for the cart to travel one meter, and compared to the actual, we were at least 0.8% off, which is almost accurate.