11/16: Moment of Inertia and Frictional Torque

Title: Moment of Inertia and Frictional Torque
Purpose: The purpose of this lab was to find the frictional torque of the apparatus and then use it to calculate and estimate how fast it would take for a cart to travel one meter.
Apparatus:

We measured the dimensions of the disk and the cylinders, and then used it to find the inertia of the system. Then, we spun the disk and used video capture to find the acceleration. With that and the inertia we found, we used it to find the frictional torque of the system. We attached the cart to the system, and let it travel down one meter and timed it. To estimate the time, we wrote equations from the rigid body diagram and then used kinematics to find the time.

Theory:
Inertia times angular acceleration should equal to the rotational torque minus the frictional torque. Since the only torque acting on the apparatus is the tension from the string, that is the only rotational torque at work. After finding the acceleration, we can plug that into a kinematics equation and find the time it takes for the cart to travel 1 meter.

Data:

Calculations: 

Calculation for moment of inertia of the apparatus

Calculation of the frictional torque

Analysis:
With all the dimensions we measured, we were able to calculate the volume. The total mass is known, but the masses of the individual parts are unknown. However, with the volume, we are able to calculate the total volume and the ratio of the volume as the whole. The mass should have the same ratio as the volume, so the individual masses could be found. With that, we could find the moments of inertia of each individual bit, and find the total inertia. That can be used to find the frictional torque, with the acceleration calculated from video capture of the spinning disk. With all of this, we found the theoretical time it would take for the cart to travel one meter while being attached to the apparatus. From the data we collected, it can be seen that the results were pretty close.

Conclusion:
The time we predicted was 10.45s, and the average of the times we measured was 10.38, with the percent error being only 0.67%, showing that the model works, and there is a frictional torque on the apparatus. The part where the most error might have occurred is during video capture to find the acceleration, where each dot may not have been at the exact correct place.