Lab 18 Moment of Inertia and Frictional Torque

Partners: Matthew Ibarra, Billy Justin
Date of Lab: 7 November 2016 

Lab 18 Moment of Inertia and Frictional Torque

Mission Statement:To predict the time of descent (the distance being 1 meter) of a cart down an angled, frictionless (presumed) slope while attached (via string) to a metal disk pulley itself subject to friction. 

Theory and Experimental Procedure for Part 1 of Lab 16: In this lab, our end goal was to determine, with as best precision as we could muster, the time it would require a cart to roll down a distance of one meter down an incline while simultaneously strung to a metal pulley. In order to carry out this calculation, we first needed to determine the moment of inertia of the pulley system when spun without the added torque applied by the cart. Since the pulley system consisted of a large disk together with cylindrical protrusions, we needed to find the respective moments and then sum them together to obtain the desired moment.

However, besides determining the moment of inertia of the system, we likewise needed to consider the opposing torque being exerted by the friction between the pulley and the supporting armature. Hence, we needed to somehow determine the angular deceleration of the pulley when spun (again, without being connected to the cart). To do this, we set up a photogate such that we could graph the theta-time graph from which we could obtain the desired angular deceleration. We taped a white strip onto the rim of the disk so that the motion could be sensed. The photogate setup is shown below. 

Using this technique, we obtained the graph shown below.

 From the graph above, we identified the angular deceleration to be twice the A value. Now equipped with this important information, we were able to proceed with our calculations.
Calculations: 
Below is our calculation for the moment of inertia of the pulley system. As aforementioned, we first found the respective moment of inertia for the disk and then the cylinder, which was broken into two slices since the disk bisected the cylinder. Once we found the component moments, we added them together and discovered our desired moment of inertia for the pulley system, which resulted in being 0.019 kg*m^2.  

Now equipped with the knowledge of the systemic moment of inertia, it was key to understand that this moment of inertia of the system, in theory, is equal to the frictional torque which eventually stops the spinning. Hence we could carry out our second calculation, shown below, in order to find the acceleration of the cart, and thus the time taken to cover a given distance. That theoretical time ended up being 7.388 seconds. 

Performing the experiment three times, we obtained three different times of 7.35, 7.32, and 7.36 seconds, all of which were approximately close to the theoretical time, and well within 4% off. Nonetheless, sources of error likely dealt with the fact that our entire calculations were based on the presumption that the pulley was uniform in terms of mass-distribution. In actuality, the pulley is not uniform, and hence the moment of inertia which we calculated is likely not the actual moment of inertia.